Linear ordinary differential equation

A fundamental system for a homogeneous linear ordinary differential equation of order

n

on an interval is a basis of its vector space of homogeneous solutions. Every solution has a unique expression

\sum_{j = 1}^{n} c_{j} y_{j}

with constant coefficients. Under the usual continuity assumptions and a nonvanishing leading coefficient, a nonzero Wronskian at one point proves that

n

solutions form such a system: their initial-value vectors are independent and span all possible initial data, and the Picard-Lindelöf theorem supplies uniqueness. For a first-order vector system, the corresponding columns form a fundamental matrix.

Fourier response of a first-order linear differential equation ( $y_{n} = g_{n} / (a + in)$ )

For

y^{'} + a y = g

with real

a > 0

and

2 π

-periodic forcing, comparing Fourier coefficients gives

(a + in) y_{n} = g_{n}

. There is no vanishing denominator, including the constant mode. With real cosine and sine coefficients

a_{n}, b_{n}

for

g

and

A_{n}, B_{n}

for

y

, the equivalent equations give

A_{n} = \frac{a a _{n} - n b _{n}}{a ^{2} + n ^{2}}, B_{n} = \frac{n a _{n} + a b _{n}}{a ^{2} + n ^{2}} .

Whenever termwise differentiation is justified, these formulas construct a periodic particular solution. For a forcing series with coefficients

O (n^{- 2})

, the differentiated response coefficients are also summable, which supplies such a justification by uniform convergence. All other solutions add the homogeneous solution

C e^{- a x}

. Since this exponential cannot be periodic unless

C = 0

, the periodic solution is unique. This is also a simple example of how a linear equation changes the amplitude and phase of each harmonic independently.

Hill's equation ( $z^{''} + k (s) z = 0$ )

A linear second-order ordinary differential equation with periodic coefficient. For accelerator beam optics,

s

measures distance along the reference orbit and

k (s)

is the local transverse focusing strength.

Oscillatory-root criterion for a real second-order equation ( $a_{1}^{2} - 4 a_{0} a_{2} < 0$ )

For real constants with

a_{2} \neq = 0

, a nonzero solution of the homogeneous linear differential equation

a_{2} f^{''} + a_{1} f^{'} + a_{0} f = 0

has a sinusoidal factor exactly when the discriminant

a_{1}^{2} - 4 a_{0} a_{2}

is negative. Indeed the characteristic polynomial has roots

- a_{1} / (2 a_{2}) \pm iω

, where

ω = 4 a_{0} a_{2} - a_{1}^{2} / (2∣ a_{2} ∣)

, giving

f = e^{- a_{1} t / (2 a_{2})} (C_{1} cos ω t + C_{2} sin ω t)

. Real or repeated roots instead give real exponential functions, possibly multiplied by a linear polynomial. Oscillatory behaviour and decay are separate: decay requires

a_{1} / a_{2} > 0

Exponential substitution removing a constant first derivative ( $y = e^{a x} u : y^{''} - 2 a y^{'} + (a^{2} + b^{2}) y = e^{a x} (u^{''} + b^{2} u)$ )

Differentiating

y = e^{a x} u

twice proves the displayed identity. It converts an exponentially weighted forced equation into a simple harmonic oscillator equation. If

b = 0

, the reduced homogeneous solutions are

1, x

rather than

cos b x, sin b x

as an independent pair. If the forcing contains a term at frequency

∣ b ∣

, resonance requires an additional polynomial factor in a particular solution.

Constant-coefficient differential equation ( $a_{n} y^{(n)} + \dots + a_{1} y^{'} + a_{0} y = f (x)$ )

A linear ordinary differential equation has constant coefficients when the multipliers

a_{j}

of the unknown function and its derivatives do not depend on the independent variable. The forcing

f

may still vary. For the homogeneous equation

f = 0

, its characteristic equation of a constant-coefficient differential equation determines exponential solutions.

Power substitution reducing a differential equation to constant coefficients ( $y = x^{n} z$ )

On an interval where the substitution is real and

x \neq = 0

, a factor formed by exponentiation can remove inverse powers from the coefficients of a linear ordinary differential equation. For

y^{''} + (a + \frac{b}{x}) y^{'} + (c + \frac{d}{x} + \frac{e}{x ^{2}}) y = 0,

the substitution

y = x^{n} z

gives

z^{''} + (a + \frac{2 n + b}{x}) z^{'} + (c + \frac{an + d}{x} + \frac{n ( n - 1 ) + bn + e}{x ^{2}}) z = 0.

It becomes a constant-coefficient differential equation exactly when

2 n + b = 0

an + d = 0

and

n (n - 1) + bn + e = 0

. Derive this by the product rule; checking only the coefficient of

z^{'}

is insufficient.

Exponential forcing response with zero initial data ( $(D - a) (D - b) y = e^{c x}, y (0) = y^{'} (0) = 0$ )

For pairwise distinct rates

a, b, c

, the response is

y = \frac{e ^{a x}}{( a - b ) ( a - c )} + \frac{e ^{b x}}{( b - a ) ( b - c )} + \frac{e ^{c x}}{( c - a ) ( c - b )} .

Set

v = y^{'} - b y

. Solving

v^{'} - a v = e^{c x}

with

v (0) = 0

gives

(e^{c x} - e^{a x}) / (c - a)

; solving the remaining first-order equation by an integrating factor gives the formula. Equivalently this is the convolution of

e^{c x}

with the zero-displacement unit-impulse response

(e^{a x} - e^{b x}) / (a - b)

on the positive half-line. In the resonant case

c = b \neq = a

, the finite limit is

y = [e^{a x} - e^{b x} - (a - b) x e^{b x}] / (a - b)^{2}

. The apparent singularity of the distinct-rate expression is removable, and the limit exhibits the polynomial factor of a resonant exponential particular solution.

Product-derivative reduction of a second-order differential equation ( $p y^{''} + 2 p^{'} y^{'} + p^{''} y = (p y)^{''}$ )

Recognizing the displayed product derivative reduces a homogeneous second-order equation to

(p y)^{''} = 0

. On any interval where

p \neq = 0

, its general solution is

y = (A + B x) / p

. For

p = 1 - x^{3}

, the equation becomes

(1 - x^{3}) y^{''} - 6 x^{2} y^{'} - 6 x y = 0

. Data

y (0) = 1

y^{'} (0) = 0

give

1/ (1 - x^{3}) = \sum_{j \geq 0} x^{3 j}

, with radius of convergence one. The poles occur at the three cube roots of unity. A solution singularity can disappear for a special numerator that cancels the corresponding factor, so an equation's singular point need not be a singularity of every solution.

Characteristic equation of a constant-coefficient differential equation ( $P (r) = 0$ )

For the homogeneous linear ordinary differential equation

\sum_{j = 0}^{n} a_{j} y^{(j)} = 0

, substitute

y = e^{r x}

. Dividing by the nonzero exponential gives the polynomial equation

P (r) = \sum_{j} a_{j} r^{j} = 0

. A simple root gives an exponential solution; a root of multiplicity

m

gives

x^{k} e^{r x}

for

0 \leq k < m

. Real equations admit real sine and cosine combinations of conjugate roots. These characteristic roots determine growth, decay and oscillation of the linearized modes.

Method of undetermined coefficients

Choose a finite trial family of polynomial-exponential or trigonometric functions for a particular solution, substitute it into a constant-coefficient linear equation, and solve for its coefficients. If the forcing exponent is a characteristic root of multiplicity

m

, multiply the ordinary trial by

t^{m}

to leave the homogeneous kernel. The method is efficient for forcing families preserved by differentiation.

Complex forcing method for a real differential equation

For a real linear differential operator

L

, one has

L (Im u) = Im (Lu)

. A forcing term written as the imaginary part of a polynomial times

e^{iω x}

can therefore be handled by solving one complex equation, then taking the imaginary part. For

L = d^{2} / d x^{2} + a

, substituting

u = p (x) e^{iω x}

gives

p^{''} + 2 iω p^{'} + (a - ω^{2}) p

. If the forcing frequency is resonant, the polynomial ansatz must include the appropriate additional power of

x

Resonant exponential particular solution ( $y_{p} = x e^{r x} / P^{'} (r)$ )

For a constant-coefficient linear differential equation

P (D) y = e^{r x}

, where

D

denotes differentiation, a simple root

P (r) = 0

forces a resonant trial. Factor

P (z) = (z - r) Q (z)

; since

(D - r) (x e^{r x}) = e^{r x}

, commuting the differential operators gives

P (D) (x e^{r x}) = Q (r) e^{r x} = P^{'} (r) e^{r x}

. Thus the displayed expression is a particular solution. More generally, if the root has multiplicity

m

, the same factorization gives

P (D) (x^{m} e^{r x}) = P^{(m)} (r) e^{r x}

, so

y_{p} = x^{m} e^{r x} / P^{(m)} (r)

. This is the extra polynomial factor required by resonance in a differential equation.

Weighted energy estimate for two coupled modes ( $N = c ∣ A ∣^{2} + a ∣ B ∣^{2}$ )

Suppose two complex amplitudes satisfy

\partial_{t} ∣ A ∣^{2} \leq 2 a ∣ A ∣∣ B ∣ - 2 d ∣ A ∣^{2}

and

\partial_{t} ∣ B ∣^{2} \leq 2 c ∣ A ∣∣ B ∣ - 2 d ∣ B ∣^{2}

, with positive

a, c

and

d \geq 0

. Then

N = c ∣ A ∣^{2} + a ∣ B ∣^{2} \geq 2 a c ∣ A ∣∣ B ∣

gives

\dot{N} \leq 2 (a c - d) N

. A positive weighted norm avoids treating two coupling rates of different size as a single arithmetic sum. This is useful for nonautonomous linear ordinary differential equations and the bounded-modulation alpha-Omega growth estimate.

Newton's law of cooling ( $y^{'} = - k (y - T_{a})$ )

Newton's law of cooling models an object's temperature

y

using a positive constant rate

k

and ambient temperature

T_{a}

. The displayed ordinary differential equation assumes one temperature represents the object, the environment remains at

T_{a}

, and the proportional rate stays fixed over the modeled range. It also describes warming when

y < T_{a}

. These are modeling assumptions: ^[ref] explain the proportional law through a tangent line approximation and caution about large temperature differences.

Dimensional analysis requires

k

to have units of inverse time. The product

k (t - t_{0})

in the exponential function solution is therefore dimensionless. The equilibrium point

y = T_{a}

attracts both warmer and colder starting states: the excess temperature undergoes exponential decay. See initial value problem for computed cooling times, direction field for the family of trajectories, and separable differential equation for a derivation.

Impulse versus finite-duration cooling ( $θ^{'} + a θ = - F (t)$ )

For equal initial temperatures and the same Newton's law of cooling coefficient

a > 0

, compare a unit Dirac delta function cooling impulse at time one with unit-rate cooling on the interval from one to two. The impulse produces a temperature jump of minus one. The finite-duration forcing gives a continuous temperature with change

- (1 - e^{- a (t - 1)}) / a

while it is active. Relative to the common unforced cooling curve, the impulse contributes

- e^{- a (t - 1)}

. Consequently the temperatures cross at

t = 1 + lo g (1 + a) / a

, strictly between one and two. After time two the impulse-cooled body is warmer: the later cooling receives less subsequent exponential attenuation.

Cumulative thermal destruction under exponential warming and cooling ( $D (T) = β_{m a x} [T - e^{- α T} (1 - e^{- α T}) / α]$ )

In an ideal linear temperature-dependent destruction model with equal warming and cooling durations, normalized temperature is

1 - e^{- α t}

during warming and

(1 - e^{- α T}) e^{- α (t - T)}

during cooling. Integrating the destruction rate gives

D (T)

, the logarithm of the surviving-count reduction factor. If the endpoint destruction rate and thermal relaxation constant stay fixed, changing the temperature gap alone leaves this integral unchanged. This is a property of the stipulated model.

Newton cooling with an instantaneous temperature jump

Under Newton's law of cooling with fixed ambient temperature

T_{0}

and

k > 0

, an instantaneous jump

β

t_{*}

sets

T (t_{*} +) = T (t_{*} -) + β

. The subsequent temperature is

T_{0} + [T (t_{*} -) + β - T_{0}] e^{- k (t - t_{*})}

. Equivalently, for an initial temperature

T_{i}

, the full solution is

T (t) = T_{0} + (T_{i} - T_{0}) e^{- k t} + β H (t - t_{*}) e^{- k (t - t_{*})} .

This piecewise model has the distributional derivative equation

T^{'} + k (T - T_{0}) = β δ (t - t_{*})

, with the Dirac delta distribution representing the added temperature.

Heat transfer coefficient ( $λ$ )

A heat transfer coefficient relates heat flux through a boundary to a temperature difference:

F = λ Δ T

. Its dimensions are power per area per temperature. In a quasistatic ice layer, atmospheric boundary resistance

1/ λ_{A}

and conductive resistance

h / k

add in series, giving effective conductance

(1/ λ_{A} + h / k)^{- 1}

Homogeneous solution

A homogeneous solution solves the associated linear equation with zero forcing. The difference of any two solutions to the same forced linear equation is homogeneous.

A homogeneous solution satisfies the associated homogeneous linear differential equation, obtained by setting the forcing term to zero. Such solutions form the kernel of the differential operator.

Particular solution

A particular solution is any one solution of a forced linear differential equation. Every solution is the sum of that particular solution and an arbitrary homogeneous solution.

General solution

The general solution of a differential equation is a family containing every solution, usually parameterized by arbitrary constants or initial data. For a forced linear ordinary differential equation, it is the sum of one particular solution and the general homogeneous solution.

Dominant eigenmode in a forced linear system

After diagonalising a constant-coefficient system, each eigenmode satisfies a scalar forced equation. The large-time behaviour is determined by the largest exponential rate whose coefficient does not vanish, including rates introduced by the forcing.

Second-order linear differential equation

A second-order linear equation has the form

y'' + p (x) y' + q (x) y = f (x)

Point interaction in a second-order differential equation ( $L_{α} y = y^{''} + α y (c) δ_{c}$ )

A point interaction concentrates a coefficient of a linear differential equation at a single interior point. On an interval

0 < x < ℓ

, with

0 < c < ℓ

, consider

L_{α} y = y^{''} + α y (c) δ_{c}

. The domain consists of continuous functions that are twice differentiable away from

c

, with the required one-sided derivatives. Multiplication of the Dirac delta by

y

means the unambiguous point value

y (c) δ_{c}

. It does not mean an undefined product of a delta with an arbitrary discontinuous function.

The equation is interpreted using distributional derivatives. Smooth forcing gives an ordinary equation on each side, with transmission conditions for a delta coefficient at the interaction. A delta potential in the one-dimensional Schrödinger equation is a physical example after choosing the corresponding sign and constants.

Dirichlet resonance for a point interaction ( $α_{*} = ℓ / (c (ℓ - c))$ )

The boundary value problem

y^{''} + α y (c) δ_{c} = f

(0, ℓ)

with Dirichlet boundary conditions has a unique solution for every continuous forcing and prescribed endpoint values exactly when

α \neq = α_{*} := ℓ / (c (ℓ - c))

. The left-normalized homogeneous solution is

u_{α} (x) = x - α c (x - c)_{+}, u_{α} (ℓ) = ℓ - α c (ℓ - c),

where

(t)_{+} = max (t, 0)

. Continuity at

c

and the transmission conditions for a delta coefficient give this formula. The final boundary condition fixes the initial slope unless the displayed endpoint value is zero. This is a one-dimensional kernel obstruction, analogous to the Fredholm alternative.

At resonance the normalized null function is

ϕ (x) = {x / c, (ℓ - x) / (ℓ - c), 0 \leq x \leq c, c \leq x \leq ℓ .

For

y (0) = a

and

y (ℓ) = b

, the compatibility condition is

\int_{0}^{ℓ} ϕ (x) f (x) d x = \frac{a}{c} + \frac{b}{ℓ - c} .

Indeed, integrate

ϕ L_{α} y - y L_{α} ϕ

on both sides of

c

. The internal terms cancel by the matching derivative jumps, leaving

[ϕ y^{'} - ϕ^{'} y]_{0}^{ℓ}

. This equals the right-hand side. The initial-slope construction also proves sufficiency: when compatible there is a one-parameter family

y_{0} + tϕ

; otherwise no solution exists.

Transmission conditions for a delta coefficient ( $[y]_{c} = 0, [y^{'}]_{c} + α y (c) = J$ )

For a point interaction in a second-order differential equation, the equation

y^{''} + α y (c) δ_{c} = f + J δ_{c}

, with nonsingular continuous

f

, requires

[y]_{c} = 0, [y^{'}]_{c} + α y (c) = J .

The distributional second derivative across a jump explains both conditions: there is no derivative-of-delta source to balance a value jump, and the coefficients of the delta source must agree. In particular, without a point source,

[y^{'}]_{c} = - α y (c)

. With a unit point source at the same location, the condition is

[y^{'}]_{c} = 1 - α y (c)

; one must combine the two contributions rather than impose two incompatible jump conditions.

Laguerre differential equation ( $x y^{''} + (1 - x) y^{'} + ν y = 0$ )

This equation has a regular singular point at zero and an irregular singular point at infinity. For a nonnegative integer

ν = n

, the solution analytic at zero with value one is the Laguerre polynomial

L_{n}

. Its weighted self-adjoint form is

(x e^{- x} y^{'})^{'} + ν e^{- x} y = 0

, connecting it to the Laguerre differential operator on polynomials.

Frobenius series for the Laguerre differential equation ( $a_{k + 1} = \frac{k - ν}{( k + 1 ) ^{2}} a_{k}$ )

A Frobenius solution

y = \sum_{k \geq 0} a_{k} x^{k + σ}

has the repeated indicial root

σ = 0

. Equating coefficients gives the displayed recurrence relation. The resulting analytic solution is an entire function: its consecutive term ratio has magnitude asymptotic to

∣ x ∣/ k

. For

ν = n \geq 0

an integer, the recurrence terminates and

L_{n} (x) = \sum_{k = 0}^{n} (- 1)^{k} (k n) x^{k} / k!

. The second local solution is independent and has a logarithmic term at zero; repeated indicial roots do not supply two independent analytic series.

Quadratic-variable reduction of a singular oscillator ( $x y^{''} - y^{'} + 4 x^{3} y = 0, u = x^{2}$ )

The chain rule with

y (x) = Y (x^{2})

gives

x y^{''} - y^{'} = 4 x^{3} Y^{''}

and transforms the equation, away from zero, into

Y^{''} + Y = 0

. Thus its solution space is spanned by

cos (x^{2})

and

sin (x^{2})

. Both extend smoothly through the regular singular point at zero. Their even power series allow the independent data

y (0)

and

y^{''} (0)

to select the two solutions; every smooth solution has

y^{'} (0) = 0

Repeated-root constant-coefficient differential equation ( $(D - r)^{2} y = f$ )

The homogeneous equation has linearly independent solutions

e^{r x}

and

x e^{r x}

, with nonzero Wronskian

e^{2 r x}

. Setting

y = e^{r x} u

converts the forced equation to

u^{''} = e^{- r x} f

. For resonant forcing

f = e^{r x}

, the particular response is

x^{2} e^{r x} /2

. This substitution explains the extra powers of

x

required by repeated-root resonance.

Singular point of a second-order linear ODE

For a normalized second-order equation, a point is singular when its coefficient functions are not both analytic there. If multiplying them by the first and second powers of distance to the point makes them analytic, it is a regular singular point. This classification concerns the equation; some individual solutions may nevertheless extend analytically through the point.

Reciprocal-coordinate reduction of a beam-type equation ( $u = t [A cos (λ / t) + B sin (λ / t)]$ )

For

t > 0

, the change of variables

τ = 1/ t

and

u (t) = v (τ) / τ

converts

t^{4} u^{''} + λ^{2} u = 0

into

v^{''} + λ^{2} v = 0

. The chain rule gives

u^{'} = v - τ v^{'}

and

u^{''} = τ^{3} v^{''}

. Thus

u = t [A cos (λ / t) + B sin (λ / t)]

. The prefactor cancels the derivative term introduced by inverting the independent variable.

Liouville transformation of a second-order equation ( $U = \overset{x}{˙}^{- 1/2} u (x (t))$ )

A smooth invertible change of variables in

u_{xx} - f (x) u = 0

introduces a first derivative in time. Rescaling by

\overset{x}{˙}^{- 1/2}

removes it. Choosing

\overset{x}{˙} = f^{- 1/2}

locally for

f > 0

produces the constant leading coefficient used in the WKB approximation; negative

f

requires a complex branch or the corresponding oscillatory real formulation.

Power substitution for an oscillatory second-order equation

For

γ \neq = 0

and

x > 0

, the change of variables

u = x^{γ}

converts

x^{2 - 2 γ} y^{''} + (1 - γ) x^{1 - 2 γ} y^{'} + γ^{2} y = 0

into

Y_{uu} + Y = 0

, where

y (x) = Y (u)

. The chain rule cancels the first-derivative terms. Thus

cos (x^{γ})

and

sin (x^{γ})

form a pair of linearly independent solutions, with Wronskian

γ x^{γ - 1}

. This is a variable-speed reparametrization of the harmonic oscillator equation.

Mathieu equation ( $u^{''} + (a - 2 q cos 2 t) u = 0$ )

The Mathieu equation is the displayed linear differential equation with periodic coefficients. Floquet theory describes its bounded and exponentially growing solutions, while its periodic Mathieu characteristic values mark the edges of parametric resonance gaps. The forcing normalization matters: replacing

t

t /2

rescales both the constant and periodic coefficients.

Mathieu characteristic value ( $a_{n} (q), b_{n} (q)$ )

The Mathieu characteristic values are the parameter values

a

for which the Mathieu equation has the even periodic mode traditionally denoted

ce_{n}

, or the odd periodic mode

se_{n}

. They are eigenvalues of

- d^{2} / d t^{2} + 2 q cos 2 t

with the corresponding periodic or antiperiodic boundary conditions. Their small-

q

expansions can be derived by projecting onto a Fourier series and perturbing the resulting tridiagonal matrix.

Airy ordinary differential equation ( $f^{''} (Z) - Z f (Z) = 0$ )

This second-order ordinary differential equation has the independent entire Airy functions

Ai

and

Bi

as a standard basis. It is distinct from the dispersive PDE in Airy equation. Its general solution is

a_{1} Ai (Z) + a_{2} Bi (Z)

Forced Airy boundary-value problem ( $- u^{''} + xu = f, u (0) = u (1) = 0$ )

On the unit interval, the weak form is

a (u, v) = \int_{0}^{1} f v

, with

a (u, v) = \int_{0}^{1} (u^{'} v^{'} + xuv)

on the zero-boundary Sobolev space. The nonnegative potential makes this form coercive in the derivative norm, and the Poincaré inequality makes that norm equivalent to the Sobolev norm. The Lax-Milgram theorem gives a unique solution. A conforming Ritz method minimizes the corresponding strictly convex quadratic energy and produces a symmetric positive-definite tridiagonal matrix for piecewise-linear hats.

Airy power-series fundamental pair ( $y_{1} (0) = 1, y_{1}^{'} (0) = 0, y_{2} (0) = 0, y_{2}^{'} (0) = 1$ )

For the Airy ordinary differential equation

y^{''} = x y

, a power series obeys

a_{2} = 0

and

a_{m + 3} = a_{m} / [(m + 3) (m + 2)]

. Setting its first two coefficients to

(1, 0)

(0, 1)

gives

y_{1} (x) = \sum_{k \geq 0} \frac{x ^{3 k}}{\prod _{j = 1}^{k} ( 3 j ) ( 3 j - 1 )}, y_{2} (x) = \sum_{k \geq 0} \frac{x ^{3 k + 1}}{\prod _{j = 1}^{k} ( 3 j ) ( 3 j + 1 )} .

Empty products are one. The ratio test proves convergence on the whole complex plane. Termwise differentiation verifies the equation, and the initial normalization gives Wronskian one. By the Abel identity, the Wronskian remains one everywhere, so these solutions span the whole solution space.

Airy function ( $Ai (Z), Bi (Z)$ )

Named after George Biddell Airy, the standard Airy functions solve the Airy ordinary differential equation. Their Wronskian is

Ai (Z) Bi^{'} (Z) - Ai^{'} (Z) Bi (Z) = 1/ π

, so they form a fundamental solution pair, including for complex

Z

Airy displacement integral ( $I_{0} (C), I_{1} (C)$ )

For

q = e^{iπ /6}

define

I_{0} (C) = \int_{0}^{\infty} Ai (q (s - C)) d s

and

I_{1} (C) = \int_{0}^{\infty} s Ai (q (s - C)) d s

. Both converge by Airy function decay. Integrating its differential equation gives

I_{1} - C I_{0} = i q Ai' (- qC)

. These moments convert a decaying vorticity profile into its velocity and displacement for three-layer long-wave shear-flow matching.

Elimination of the first derivative

For a second-order linear differential equation

y^{''} + p (x) y^{'} + q (x) y = 0,

the substitution

y = u exp [- \frac{1}{2} \int p (x) d x]

eliminates the first-derivative term and gives

u^{''} + (q - \frac{1}{2} p^{'} - \frac{1}{4} p^{2}) u = 0.

Sturm-Picone comparison theorem

For two real self-adjoint equations

(p_{i} y_{i}')' + q_{i} y_{i} = 0

with continuous coefficients,

0 < p_{2} \leq p_{1}

and

q_{1} \leq q_{2}

, between consecutive zeros of a nontrivial first solution a nontrivial second solution either has a zero or is proportional to the first there. This comparison controls oscillation through the coefficients.

Sturm comparison theorem

q_{1} \leq q_{2}

, then between consecutive zeros of a nontrivial solution of

y^{''} + q_{1} (x) y = 0

there is a zero of every nontrivial solution of

y^{''} + q_{2} (x) y = 0

, unless the two coefficients agree throughout that interval.

Wronskian proof of Sturm comparison

For solutions

φ_{i}^{''} + q_{i} φ_{i} = 0

, the mixed Wronskian

W = φ_{1}^{'} φ_{2} - φ_{1} φ_{2}^{'}

satisfies

W^{'} = (q_{2} - q_{1}) φ_{1} φ_{2} .

Its endpoint signs between consecutive zeros of a positive

φ_{1}

force a zero of

φ_{2}

unless

q_{1} = q_{2}

Power-series solution of a differential equation

Kummer differential equation

Kummer's equation is

x y^{''} + (b - x) y^{'} - a y = 0.

Its solution analytic at zero is the confluent hypergeometric function of the first kind

M (a, b, x) = \sum_{m = 0}^{\infty} \frac{( a ) _{m}}{( b ) _{m} m !} x^{m} .

When

b

is not an integer, a second local solution is

x^{1 - b} M (a - b + 1, 2 - b, x) .

Exponential reduction of a confluent hypergeometric equation with equal parameters ( $z Y^{''} + (b - z) Y^{'} - bY = 0$ )

Putting

Y = e^{z} V

in the displayed Kummer differential equation gives

z V^{''} + (z + b) V^{'} = 0

. Thus

V^{'} = C z^{- b} e^{- z}

on a chosen branch, yielding

Y = e^{z} [A + B \int z^{- b} e^{- z} d z]

. For

b = 1/2

and

z = x /2

, one solution is

e^{x /2}

and a normalized second Frobenius solution is

x \sum_{n \geq 0} x^{n} / (2 n + 1)!!

. The analytic factors in both solutions have infinite radius of convergence, since their coefficient ratios tend to zero. The second solution still has a square-root branch at zero; an entire series factor does not remove a nonintegral Frobenius exponent.

Confluent hypergeometric function

Confluent hypergeometric functions solve the Kummer differential equation

z y^{''} + (b - z) y^{'} - a y = 0

. The regular solution is the confluent hypergeometric function of the first kind; a second standard solution is conventionally denoted

U (a, b, z)

. The family arises by coalescing singular points of the Gauss hypergeometric equation.

Confluent hypergeometric function of the first kind ( $M (a, b, z) =_{1} F_{1} (a; b; z)$ )

For

b

not a nonpositive integer, the regular solution normalized by

M (a, b, 0) = 1

has the power series

M (a, b, z) = \sum_{n = 0}^{\infty} \frac{( a ) _{n}}{( b ) _{n}} \frac{z ^{n}}{n !} .

Here

(a)_{n}

is the rising factorial. Substitution into the Kummer differential equation proves the coefficient recurrence. This normalization agrees with NIST's definitions.

Wronskian ( $W (f, g) = f g^{'} - f^{'} g$ )

For two differentiable functions

f, g

on an interval

I

, their Wronskian is the function

W (f, g) (x) = det (f (x) f^{'} (x) g (x) g^{'} (x)) = f (x) g^{'} (x) - f^{'} (x) g (x) .

The order matters: exchanging

f

and

g

reverses the sign. For

n

sufficiently differentiable functions, the Wronskian is the determinant

det (f_{j}^{(i - 1)} (x))_{i, j = 1}^{n}

.^[ref]

The Wronskian measures whether the vectors

(f (x), f^{'} (x))

and

(g (x), g^{'} (x))

supply independent value-and-derivative data at that point. For real-valued functions, its absolute value is the area of the parallelogram they span. If a constant linear combination

a f + b g

is zero throughout

I

, then

a f^{'} + b g^{'} = 0

too. A nonzero Wronskian at even one point makes this pair of linear equations invertible and forces

a = b = 0

. Thus a nonzero Wronskian somewhere always proves linear independence of the functions on

I

For an explicit example, the exponential functions

y_{1} (x) = e^{- x}, y_{2} (x) = e^{2 x}

solve the same homogeneous linear differential equation

y^{''} - y^{'} - 2 y = 0

on the whole real line: substitution gives

1 + 1 - 2 = 0

and

4 - 2 - 2 = 0

, respectively. Their Wronskian is

W (y_{1}, y_{2}) = e^{- x} (2 e^{2 x}) - (- e^{- x}) e^{2 x} = 3 e^{x} > 0.

For the initial value problem

y (0) = 2

y^{'} (0) = - 1

, the linear combination

y = c_{1} y_{1} + c_{2} y_{2}

requires

c_{1} + c_{2} = 2

and

- c_{1} + 2 c_{2} = - 1

. Hence

c_{1} = 5/3

c_{2} = 1/3

, and

y = \frac{5}{3} e^{- x} + \frac{1}{3} e^{2 x}

. The nonzero Wronskian makes the two initial conditions independently adjustable.

Figure 1.
Independent exponential functions supply independent initial data, with Wronskian equal to the oriented parallelogram area
. **Details**.

The stronger converse requires a common equation. Suppose

y_{1}, y_{2}

solve

y^{''} + p (x) y^{'} + q (x) y = 0

with continuous functions

p, q

on one interval

I

. Then the functions are linearly independent if and only if their Wronskian is nonzero at one point, equivalently at every point. If

W (x_{0}) = 0

, choose constants

a, b

, not both zero, making

z = a y_{1} + b y_{2}

satisfy

z (x_{0}) = z^{'} (x_{0}) = 0

. The superposition principle makes

z

a solution of a differential equation for the same equation, and the uniqueness theorem for ordinary differential equations forces

z

to be identically zero. This proves linear dependence.^[ref]

For arbitrary differentiable functions, an identically zero Wronskian need not imply linear dependence. Consider, on

I = (- 2, 2)

f (x) = x^{4}, g (x) = x ∣ x ∣^{3} = {- x^{4}, x^{4}, x < 0, x \geq 0.

(4)

Both functions have three continuous derivatives. On each side of zero,

g

is a constant multiple of

f

, so

W (f, g) = 0

; at zero their values and first derivatives are all zero, so the Wronskian is zero there too. Nevertheless

a f + b g = 0

evaluated at

x = - 1

gives

a - b = 0

, while evaluation at

x = 1

gives

a + b = 0

. Hence

a = b = 0

: these functions are linearly independent on the whole interval. There is no single constant multiplier valid on both sides of the common zero.

Figure 2.
An identically zero Wronskian can coexist with linear independence when the common regular equation hypothesis is absent
. **Details**.

These functions cannot both solve any equation

y^{''} + p y^{'} + q y = 0

with continuous functions

p, q

throughout

(- 2, 2)

: even

f

alone would contradict uniqueness because

f (0) = f^{'} (0) = 0

but

f

is not identically zero. They do solve the singular equation

x^{2} y^{''} - 3 x y^{'} = 0

, whose leading coefficient vanishes at zero. Dividing by

x^{2}

gives

p = - 3/ x

only away from zero; Abel's identity cannot be applied across that point. The example pinpoints why checking the equation and its interval is essential before using a zero Wronskian to infer linear dependence.

Vanishing Wronskian does not imply linear dependence

Two differentiable functions can be linearly independent on a connected interval while their Wronskian vanishes everywhere. For example, let

f (x) = (max {- x, 0})^{2}

and

g (x) = (max {x, 0})^{2}

. Both have a continuous first derivative, and their values and first derivatives vanish at zero. On either side of zero one of the functions is identically zero, so

f g^{'} - f^{'} g = 0

everywhere. Evaluating a constant linear combination

A f + B g = 0

- 1

and

1

forces

A = B = 0

The obstruction is that the two halves supply independent information without simultaneous nonzero value-and-derivative data at any point. For solutions of a common regular second-order linear differential equation, the uniqueness theorem for ordinary differential equations removes this possibility; Abel's identity gives the zero-or-nowhere-zero alternative for their Wronskian.

Abel's identity ( $W (x) = W (x_{0}) exp (- \int_{x_{0}}^{x} p (s) d s)$ )

For

y'' + p y' + q y = 0

, the Wronskian satisfies

W' = - p W

More precisely, let

y_{1}, y_{2}

be solutions of a differential equation for the same homogeneous linear differential equation

y^{''} + p (x) y^{'} + q (x) y = 0

on an interval

I

where

p, q

are continuous functions. For any

x_{0} \in I

, Abel's identity gives

W (y_{1}, y_{2}) (x) = W (y_{1}, y_{2}) (x_{0}) exp (- \int_{x_{0}}^{x} p (s) d s) .

Only the coefficient of

y^{'}

enters the evolution of the Wronskian. In an unnormalized equation

A (x) y^{''} + B (x) y^{'} + C (x) y = 0

, first use

p = B / A

on an interval where

A \neq = 0

and the coefficient ratios are continuous functions.^[ref]

The proof uses cancellation. Differentiating the Wronskian by the product rule and substituting the common equation gives

W^{'} = (y_{1} y_{2}^{'} - y_{1}^{'} y_{2})^{'} = y_{1} y_{2}^{''} - y_{1}^{''} y_{2} = y_{1} (- p y_{2}^{'} - q y_{2}) - (- p y_{1}^{'} - q y_{1}) y_{2} = - p W .

Multiplying this first-order linear differential equation by the integrating factor

μ (x) = exp (\int_{x_{0}}^{x} p (s) d s)

gives

(μ W)^{'} = 0

. Since

μ (x_{0}) = 1

, the constant is

W (x_{0})

, proving Abel's identity. This argument also covers

W (x_{0}) = 0

without dividing by

W

The exponential function is positive for real

p

, so a real Wronskian either vanishes everywhere or has a fixed nonzero sign throughout

I

. In the preceding example,

p = - 1

and

W (0) = 3

give

W (x) = 3 e^{x}

immediately. If

p = 0

, every Wronskian of two solutions of a differential equation for that same equation is constant, regardless of

q

. Thus Abel's identity can certify linear independence and find the denominator used in variation of parameters without explicit formulas for both solutions of a differential equation.

Variation of parameters ( $y_{p} = - y_{1} \int y_{2} r / W + y_{2} \int y_{1} r / W$ )

Variation of parameters replaces the constants in a complementary solution by functions to construct a particular solution.

For the inhomogeneous linear differential equation

y^{''} + p y^{'} + q y = r

, assume

p, q, r

are continuous functions on an interval

I

, and

y_{1}, y_{2}

are linearly independent homogeneous solutions there. With

W = W (y_{1}, y_{2})

and

x_{0} \in I

, variation of parameters constructs the particular solution

y_{p} (x) = - y_{1} (x) \int_{x_{0}}^{x} \frac{y _{2} ( s ) r ( s )}{W ( s )} d s + y_{2} (x) \int_{x_{0}}^{x} \frac{y _{1} ( s ) r ( s )}{W ( s )} d s .

The Wronskian has no zeros on

I

, so these quotients are defined. For

A y^{''} + B y^{'} + C y = F

, normalize first:

r = F / A

, not

F

.^[ref]

To derive the formula, write

y_{p} = u_{1} y_{1} + u_{2} y_{2}

and impose

u_{1}^{'} y_{1} + u_{2}^{'} y_{2} = 0

. This auxiliary choice removes extra terms from the first derivative. Substitution into the inhomogeneous linear differential equation leaves

u_{1}^{'} y_{1}^{'} + u_{2}^{'} y_{2}^{'} = r

. The two linear equations for

u_{1}^{'}, u_{2}^{'}

have determinant

W

, and Cramer's rule gives

u_{1}^{'} = - \frac{y _{2} r}{W}, u_{2}^{'} = \frac{y _{1} r}{W} .

Integrating from

x_{0}

makes

y_{p} (x_{0}) = y_{p}^{'} (x_{0}) = 0

. Arbitrary integration constants contribute only homogeneous solutions; adding

c_{1} y_{1} + c_{2} y_{2}

supplies the general solution.

For a worked example, use the same homogeneous solutions

e^{- x}, e^{2 x}

and the forcing term

r = 6 e^{3 x}

y^{''} - y^{'} - 2 y = 6 e^{3 x}

. With

x_{0} = 0

and

W = 3 e^{x}

u_{1}^{'} = - 2 e^{4 x}, u_{2}^{'} = 2 e^{x}, u_{1} = - \frac{1}{2} (e^{4 x} - 1), u_{2} = 2 (e^{x} - 1) .

Consequently the particular solution selected by zero initial conditions is

y_{p} (x) = \frac{3}{2} e^{3 x} + \frac{1}{2} e^{- x} - 2 e^{2 x} .

(4)

Its value and first derivative vanish at zero. Direct substitution confirms the forcing term: the differential operator

L [y] = y^{''} - y^{'} - 2 y

annihilates the last two terms and maps

e^{3 x}

4 e^{3 x}

. The two integrals therefore build a response from known homogeneous solutions, even when the forcing term has no convenient elementary guess.

Reduction of order

Given one nonzero solution of a homogeneous second-order linear equation, reduction of order constructs a second by writing it as a variable multiple of the first.

First correction to a regular singular solution by reduction of order ( $y_{2} = x^{P_{0} + 1} + \frac{( P _{0} + 1 ) P _{1} - P _{0} β}{P _{0} + 2} x^{P_{0} + 2} + \dots$ )

Suppose

y^{''} - P (x) y^{'} / x - Q (x) y / x = 0

has analytic

P, Q

near zero,

P (x) = P_{0} + P_{1} x + \dots

, and a solution

y_{1} = 1 + β x + \dots

, with

P_{0}

a positive integer. The indicial roots are

0, P_{0} + 1

. The Abel identity gives

W = C x^{P_{0}} (1 + P_{1} x + \dots)

. Dividing by

y_{1}^{2}

gives

C x^{P_{0}} [1 + (P_{1} - 2 β) x + \dots]

. Integrating this reduction of order expression from zero, multiplying by

y_{1}

, and choosing

C = P_{0} + 1

proves the displayed normalized expansion. The integral's zero constant eliminates the added multiple of

y_{1}

. Its nonzero Wronskian proves independence. In this setting the assumed exponent-zero solution prevents a logarithm in the constructed exponent-

P_{0} + 1

solution.

Reduction of order across a zero of the known solution

The integral representation

y_{2} = y_{1} \int e^{- \int p} / y_{1}^{2}

is derived where

y_{1} \neq = 0

. At a zero, its integral factor may have a pole while the product has a removable singularity. For regular coefficients, continue the actual second solution by its finite initial data and initial-value uniqueness; the Abel identity preserves a nonzero Wronskian. In the degree-one Legendre differential equation,

y_{1} = x

and

y_{2} = x arctanh x - 1

illustrate this cancellation at zero.

Cauchy-Euler equation

A Cauchy–Euler equation has powers of the independent variable matched to derivative order and is solved using power laws or a logarithmic change of variable.

Logarithmic conjugacy of a repeated-root Euler operator ( $x^{2} D^{2} - x D + 1 = x (d^{2} / d t^{2}) x^{- 1}, t = lo g x$ )

For

x > 0

, write

t = lo g x

and

y = xv (t)

. The chain rule gives

y^{'} = v + v_{t}

y^{''} = x^{- 1} (v_{t} + v_{tt})

, and therefore

(x^{2} D^{2} - x D + 1) y = x v_{tt} .

1 < x < e^{ℓ}

with homogeneous Dirichlet boundary conditions, the eigenfunctions are

x sin (nπ lo g x / ℓ)

and the eigenvalues are

- (nπ / ℓ)^{2}

. The weighted inner product with weight

x^{- 3}

becomes

\int_{0}^{ℓ} \overline{v} w d t

, so their squared norms are

ℓ /2

and their orthogonality is exactly that of the Fourier sine series. The weight is the factor that makes this conjugacy preserve the inner product.

Logarithmic change of variable in a Sturm-Liouville problem ( $t = lo g x, x^{2} y^{''} + x y^{'} = Y^{''} (t)$ )

For a Cauchy-Euler equation, logarithmic coordinates turn scale changes in

x > 0

into translations in

t

. With

Y (t) = y (e^{t})

, the chain rule gives

x y^{'} = Y^{'}

and

x^{2} y^{''} = Y^{''} - Y^{'}

. Consequently the Sturm-Liouville operator

- x^{2} y^{''} - x y^{'} - cy

becomes

- Y^{''} - c Y

. Its natural weighted inner product has measure

d x / x = d t

, so this change of variable also preserves the norm when the transformed problem uses ordinary integration in

t

. On

1 < x < e^{ℓ}

with Dirichlet boundary conditions, the orthonormal eigenfunctions are

2/ ℓ sin (nπ lo g x / ℓ)

and the eigenvalues are

(nπ / ℓ)^{2} - c

, for

n \geq 1

Dirichlet spectrum with inverse-square weight ( $λ_{n} = 1/4 + (nπ / ℓ)^{2}, y_{n} = 2/ ℓ x sin (nπ ln x / ℓ)$ )

For

- y^{''} = λ x^{- 2} y

1 < x < e^{ℓ}

with zero endpoint values, the natural weighted inner product has measure

d x / x^{2}

. Set

v = ln x

and

y (x) = x Z (v)

. The chain rule gives

x^{2} y^{''} = x (Z^{''} - Z /4)

, so

- Z^{''} + Z /4 = λ Z

with zero endpoints on

(0, ℓ)

. Moreover

∣ y ∣^{2} d x / x^{2} = ∣ Z ∣^{2} d v

, so the transformation preserves the norm. The ordinary sine orthonormal eigenbasis therefore gives exactly the displayed spectrum and eigenfunctions for

n \geq 1

. The factor

x

and the additive

1/4

distinguish this problem from the operator

- x^{2} y^{''} - x y^{'}

with measure

d x / x

Constant forcing in a logarithmic Dirichlet Sturm-Liouville problem ( $- x^{2} y^{''} - x y^{'} - α y = 1, y (1) = y (e^{ℓ}) = 0$ )

For

ℓ > 0

, the logarithmic change of variable in a Sturm-Liouville problem sends this equation to

- Y^{''} - α Y = 1

(0, ℓ)

. The normalized eigenfunctions are

2/ ℓ sin (nπ t / ℓ)

, with eigenvalues

λ_{n} = (nπ / ℓ)^{2} - α

. If every

λ_{n}

is nonzero, the Sturm-Liouville eigenfunction expansion gives

y (x) = n \geq 1 n odd \sum \frac{4 sin ( nπ lo g x / ℓ )}{nπ [( nπ / ℓ ) ^{2} - α ]} .

Indeed the constant function has sine coefficients

2 [1 - (- 1)^{n}] / (nπ)

; division by each eigenvalue solves the equation in the operator domain. The series is uniformly convergent, with coefficients

O (n^{- 3})

, while applying the differential operator is justified in the weighted square-integrable sense. At a zero eigenvalue the solvability condition at a Sturm-Liouville eigenvalue must instead be checked: an odd resonant index makes the constant forcing incompatible, whereas an even resonant index permits solutions with an arbitrary multiple of that zero mode.

Euler-Cauchy classification from two regular singular points ( $z^{2} y^{''} + A z y^{'} + B y = 0$ )

Normalize the second-order equation as

y^{''} + P (z) y^{'} + Q (z) y = 0

. If every point other than zero and infinity is ordinary and both exceptions are at worst regular singular points, then

z P

and

z^{2} Q

extend holomorphically across zero and stay bounded at infinity. Liouville's theorem makes them constants, yielding the displayed Euler-Cauchy equation. If both exceptions must be genuinely singular, exclude

(A, B) = (0, 0)

and

(2, 0)

, which make zero and infinity respectively ordinary.

Euler-Cauchy equation

General solution of an Euler-Cauchy equation

For

z^{2} w^{''} + a z w^{'} + b w = 0,

the power ansatz

w = z^{λ}

gives the indicial equation

λ (λ - 1) + aλ + b = 0

. Distinct roots give

C_{1} z^{λ_{1}} + C_{2} z^{λ_{2}}

; a repeated root

λ

gives

z^{λ} (C_{1} + C_{2} lo g z)

on a chosen logarithm branch.

Power-law ansatz ( $y = x^{s}$ )

A power-law ansatz assumes that an unknown function is a monomial

y = x^{s}

. Scale-invariant differential equations then reduce to an algebraic equation for the exponent

s

Indicial equation

An indicial equation is the algebraic equation for the leading exponent of a power-law or Frobenius solution near a regular singular point.

Indicial root ( $P (p) = 0$ )

An indicial root is a root of the indicial equation obtained from a power-law ansatz. For a Cauchy-Euler differential equation, such roots give the exponents of its power solutions. Their sign determines which modes remain bounded near a singular endpoint.

Indicial exponent ( $r$ )

An indicial exponent is a root of the indicial equation at a regular singular point. It gives the leading local power in a Frobenius method; resonant roots can require logarithmic terms.

Log-periodic oscillation

A log-periodic oscillation is periodic in the logarithm of its argument, for example

cos (μ lo g x)

. Its successive zeros or extrema therefore occur at radii in a geometric progression.

Legendre differential equation

The equation

(1 - x^{2}) y'' - 2 x y' + ℓ (ℓ + 1) y = 0

has polynomial solutions for nonnegative integer

ℓ

The equation

(1 - x^{2}) y'' - 2 x y' + ℓ (ℓ + 1) y = 0

has the Legendre polynomial of degree

ℓ

as its normalized regular polynomial solution when

ℓ

is a nonnegative integer.

Frobenius expansion of the Legendre equation at an endpoint ( $a_{k + 1} = \frac{( k - n ) ( k + n + 1 )}{2 ( k + 1 ) ^{2}} a_{k}$ )

For the Legendre differential equation

(1 - x^{2}) y^{''} - 2 x y^{'} + n (n + 1) y = 0

, put

t = 1 + x

. The equation becomes

t (2 - t) y_{tt} + 2 (1 - t) y_{t} + n (n + 1) y = 0

. Its indicial equation is

2 r^{2} = 0

. The exponent-zero power series

\sum a_{k} t^{k}

obeys the displayed linear recurrence relation. For a nonnegative integer

n

, it terminates at degree

n

, giving a Legendre polynomial with

a_{0} \neq = 0

. Every analytic solution at the endpoint is proportional to this one, since the recurrence determines all coefficients from

a_{0}

The second independent solution has the local form

P_{n} (x) lo g (1 + x) + B (x)

, where

B

is analytic near

- 1

. Indeed, reduction of order gives

P_{n} (x) \int d x / [(1 - x^{2}) P_{n} (x)^{2}]

, whose integrand is

1/ [2 P_{n} (- 1)^{2} (1 + x)]

plus an analytic function. Rescale the second solution to obtain the unit coefficient of

P_{n} lo g (1 + x)

. Reflection gives the corresponding expansion at

x = 1

. This also proves the logarithmic divergence of the independent Legendre function of the second kind at each endpoint.

Frobenius recurrence for the Legendre equation ( $a_{n + 2} = \frac{( n + σ ) ( n + σ + 1 ) - λ ( λ + 1 )}{( n + σ + 2 ) ( n + σ + 1 )} a_{n}$ )

Substitute

y = x^{σ} \sum_{n \geq 0} a_{n} x^{n}

a_{0} \neq = 0

, into the Legendre differential equation. The lowest powers give

σ (σ - 1) = 0

and

σ (σ + 1) a_{1} = 0

; subsequent powers give the recurrence in the title. Thus exponent zero has independent even and odd coefficients, while exponent one requires

a_{1} = 0

and represents the odd solution. Zero is an ordinary point of the equation, so the exponent-one choice simply selects a solution that vanishes there.

The singular points at

\pm 1

bound the generic radius of convergence about zero by one. Every nonterminating parity recurrence has successive-coefficient ratio tending to one in the variable

x^{2}

, giving radius exactly one. If

λ = N

is a nonnegative integer, choosing the parity of

N

makes the recurrence terminate at degree

N

, producing a Legendre polynomial with infinite radius. These are precisely the nonnegative parameters with a nonzero entire solution: even and odd parts of an entire solution would both have to be entire, so any nonzero nonterminating part is excluded.

Legendre function of the second kind ( $Q_{n} (x)$ )

For nonnegative integer

n

, a second solution of the Legendre differential equation is obtained by reduction of order from

P_{n}

P_{n} (x) \int d x / [(1 - x^{2}) P_{n} (x)^{2}]

. Normalize it as

Q_{0} = \frac{1}{2} lo g ∣ (1 + x) / (1 - x) ∣

and

Q_{1} = x Q_{0} - 1

, with the usual Legendre polynomial recurrence relation for higher degrees. Inside

(- 1, 1)

Q_{n} \sim - \frac{1}{2} lo g (1 - x)

near

1

and

Q_{n} \sim (- 1)^{n} \frac{1}{2} lo g (1 + x)

near

- 1

. These follow from

P_{n} (1) = 1

P_{n} (- 1) = (- 1)^{n}

in the integral. Poles in the integral at interior polynomial zeros cancel in the product, by reduction of order across a zero of the known solution. On

x > 1

, the integral from infinity defines the decaying normalization

Q_{n} (x) \sim 2^{n} (n!)^{2} x^{- n - 1} / (2 n + 1)!

. For

n \geq 1

all bounded exterior solutions decay this way; for

n = 0

an additional constant solution remains bounded.

Associated Legendre differential equation ( $[(1 - x^{2}) y^{'}]^{'} + [ℓ (ℓ + 1) - m^{2} / (1 - x^{2})] y = 0$ )

This second-order linear differential equation extends the Legendre differential equation by its order parameter

m

. Its solutions are associated Legendre functions. In a degree-one potential problem, with

x = tanh η

, two explicit solutions are

e^{m η} (m - tanh η)

and

e^{- m η} (m + tanh η)

. Their Wronskian with respect to

η

2 m (1 - m^{2})

, so the two formulas are not a fundamental pair at

m = 0

m = \pm 1

Associated Legendre function ( $P_{ν}^{μ} (x), Q_{ν}^{μ} (x)$ )

Associated Legendre functions solve

(1 - x^{2}) y^{''} - 2 x y^{'} + [ν (ν + 1) - μ^{2} / (1 - x^{2})] y = 0

. For integers

ℓ \geq 0

and

0 \leq m \leq ℓ

, the regular solution is

P_{ℓ}^{m} (x) = (- 1)^{m} (1 - x^{2})^{m /2} (d / d x)^{m} P_{ℓ} (x)

, using a Legendre polynomial. It enters normalized spherical harmonics. Despite the traditional term associated Legendre polynomial, the factor

(1 - x^{2})^{m /2}

is not a polynomial when

m

is odd. Negative integer orders satisfy

P_{ℓ}^{- m} = (- 1)^{m} (ℓ - m)! P_{ℓ}^{m} / (ℓ + m)!

Legendre polynomial ( $P_{l} (x)$ )

The polynomial solution of polynomial degree

l

normalized by

P_{l} (1) = 1

is the Legendre polynomial

P_{l}

Legendre polynomial evaluation representer ( $q_{n} (t) = \sum_{j = 0}^{n} \overline{p_{j} (0)} p_{j} (t)$ )

For orthonormal Legendre polynomials on

[- 1, 1]

, the polynomial

q_{n}

represents evaluation at zero on polynomials of degree at most

n

⟨ r, q_{n} ⟩ = r (0)

. This follows by expanding

r

in the orthonormal basis, or from the Riesz representation theorem. With

M = ⌊ n /2 ⌋

and

c_{m} = (m 2 m) / 4^{m}

, orthonormality gives

∥ q_{n} ∥_{2}^{2} = \frac{1}{2} \sum_{m = 0}^{M} (4 m + 1) c_{m}^{2} = \frac{( 2 M + 1 ) ^{2} c _{M}^{2}}{2} .

The last identity telescopes using

c_{m} / c_{m - 1} = (2 m - 1) / (2 m)

. The norms are unbounded:

c_{m} \geq 1/ (2 m + 1)

makes the sum dominate a divergent harmonic series. Thus the evaluation identities hold eventually for every fixed polynomial but cannot extend to convergence for every

L^{2}

vector, by the Uniform boundedness principle. Point evaluation is also not defined on

L^{2}

equivalence classes.

Legendre expansion of the sign function ( $⟨ sgn, p_{2 m + 1} ⟩ = (- 1)^{m} (4 m + 3) /2 (2 m)! / [2^{2 m} m! (m + 1)!]$ )

For normalized Legendre polynomials

p_{n} = (2 n + 1) /2 P_{n}

, parity makes the sign function orthogonal to every even index. Integrating the Legendre differential equation gives

\int_{0}^{1} P_{n} = P_{n}^{'} (0) / (n (n + 1))

. The differentiated generating function

z (1 + z^{2})^{- 3/2}

gives the derivatives at zero and hence the displayed odd coefficients. The Parseval identity and squared norm two then evaluate

\sum_{m \geq 0} ((2 m)!)^{2} (4 m + 3) / [2^{4 m} (m! (m + 1)!)^{2}] = 4

Rodrigues formula for Legendre polynomials ( $P_{l} (x) = \frac{1}{2 ^{l} l !} \frac{d ^{l}}{d x ^{l}} (x^{2} - 1)^{l}$ )

This instance of Rodrigues' formula constructs the Legendre polynomial for every nonnegative integer

l

. Put

v = (x^{2} - 1)^{l}

; then

(x^{2} - 1) v^{'} = 2 l xv

. Differentiating

l + 1

times by the Leibniz rule gives

(x^{2} - 1) y^{''} + 2 x y^{'} - l (l + 1) y = 0

for

y = v^{(l)}

, which is the Legendre differential equation after reversing signs. The leading term of the normalized derivative is

(2 l)! x^{l} / [2^{l} (l!)^{2}]

, proving its polynomial degree is exactly

l

. At

x = 1

, the product

(x + 1)^{l} (x - 1)^{l}

contributes only the term with all

l

derivatives on the second factor, giving

v^{(l)} (1) = 2^{l} l!

. Thus the normalization is

P_{l} (1) = 1

Weighted orthogonality of Legendre polynomial derivatives ( $\int_{- 1}^{1} (1 - x^{2}) P_{n}^{'} (x) P_{m}^{'} (x) d x = 0 (n \neq = m)$ )

Differentiating the Legendre differential equation gives

((1 - x^{2})^{2} R_{n}^{'})^{'} + (n - 1) (n + 2) (1 - x^{2}) R_{n} = 0

for

R_{n} = P_{n}^{'}

. Multiply the equations for two distinct indices by the opposite polynomial and subtract. The endpoint term vanishes because of

(1 - x^{2})^{2}

, giving the displayed weighted orthogonality. The zero polynomial

R_{0}

is not a nonzero eigenfunction.

Generating function and norm of Legendre polynomials ( $(1 - 2 x t + x^{2})^{- 1/2} = \sum_{n \geq 0} x^{n} P_{n} (t)$ )

With

P_{n} (1) = 1

, the generating function satisfies

\partial_{x} (x^{2} \partial_{x} G) + \partial_{t} ((1 - t^{2}) \partial_{t} G) = 0

. Expanding in Legendre polynomials gives

x^{2} a_{n}^{''} + 2 x a_{n}^{'} - n (n + 1) a_{n} = 0

. Boundedness at zero removes

x^{- n - 1}

, and the value at

t = 1

makes

a_{n} = x^{n}

. Orthogonality then gives

\int_{- 1}^{1} G^{2} d t = \sum_{n} k_{n} x^{2 n}

. The integral is

x^{- 1} lo g ((1 + x) / (1 - x))

, so

k_{n} = 2/ (2 n + 1)

. Uniform analyticity on compact subdiscs

∣ x ∣ < 1

justifies the generating expansion and termwise calculations.

Orthogonality of Legendre polynomials

The Legendre polynomials are orthogonal on

[- 1, 1]

with unit weight:

\int_{- 1}^{1} P_{ℓ} (x) P_{m} (x) d x = \frac{2}{2 ℓ + 1} δ_{ℓ m} .

Legendre polynomial recurrence relation

The three-term recurrence

(2 ℓ + 1) x P_{ℓ} (x) = (ℓ + 1) P_{ℓ + 1} (x) + ℓ P_{ℓ - 1} (x)

turns multiplication by

x

into coupling between adjacent Legendre multipoles.

Schläfli contour integral for Legendre polynomials

For a contour enclosing

t

P_{n} (t) = \frac{1}{2 ^{n + 1} πi} \oint \frac{( z ^{2} - 1 ) ^{n}}{( z - t ) ^{n + 1}} d z .

Debye asymptotic for Legendre polynomials

For fixed

0 < θ < π

P_{n} (cos θ) \sim \frac{2}{πn s i n θ} cos ((n + \frac{1}{2}) θ - \frac{π}{4}) .

Hyperbolic Debye asymptotic for Legendre polynomials

For fixed

ξ > 0

P_{n} (cosh ξ) \sim \frac{e ^{(n + 1/2) ξ}}{2 πn s i n h ξ} .

Legendre boundary layer at x equals one ( $P_{n} (1 + ν / n^{2}) \to I_{0} (2 ν)$ )

The endpoint Laplace method for a Legendre polynomial ceases to localize when

n x - 1 = O (1)

. On the scale

x = 1 + ν / n^{2}

, its real integral representation gives

P_{n} (1 + ν / n^{2}) ⟶ \frac{1}{π} \int_{0}^{π} e^{2 ν c o s θ} d θ = I_{0} (2 ν) .

The Modified Bessel function of the first kind equals one at

ν = 0

and has large-

ν

behavior

e^{2 ν} / 2 π 2 ν

, matching the Hyperbolic Debye asymptotic for Legendre polynomials. This boundary layer resolves an increasingly steep function at an endpoint where its value remains fixed.

Change of independent variable in a second-order ODE

For

z = z (x)

, the second derivative transforms as

w_{xx} = z_{x}^{2} w_{zz} + z_{xx} w_{z}

First-order linear differential equation ( $y^{'} + p (x) y = q (x)$ )

A first-order linear differential equation has the form

y^{'} + p (x) y = q (x)

. Multiplication by an integrating factor

μ

satisfying

μ^{'} = p μ

turns it into

(μ y)^{'} = μ q

Finite-pulse response of a first-order linear equation ( $y (t) = A [\frac{e ^{λ (t - a)} - 1}{λ} H (t - a) - \frac{e ^{λ (t - b)} - 1}{λ} H (t - b)]$ )

Let

0 < a < b

and

y^{'} - λ y = A [H (t - a) - H (t - b)]

, with zero initial data. For

λ \neq = 0

, the Laplace transform time-shift rule gives the displayed response, since the transform of the input is

A (e^{- a s} - e^{- b s}) / s

and the equation supplies the factor

(s - λ)^{- 1}

. Equivalently, integrate the impulse response

e^{λ t} H (t)

over the active pulse. Before

a

the solution is zero; during the pulse it is

A (e^{λ (t - a)} - 1) / λ

; after

b

it is

A e^{λ t} (e^{- λa} - e^{- λb}) / λ

. The solution is continuous at both switching times. At

λ = 0

take the limit, giving linear growth during the pulse and the constant

A (b - a)

afterward. Positive

λ

gives growing memory, while negative

λ

gives decay after forcing stops.

Causal response of a first-order relaxation equation ( $G (t) = α^{- 1} e^{- t / α} H (t)$ )

For

α > 0

, the retarded impulse response of

α y^{'} + y = f

is the displayed function, with

H

the Heaviside step function. The corresponding convolution

y (t) = α^{- 1} \int_{- \infty}^{t} e^{- (t - s) / α} f (s) d s

is the unique solution tending to zero in the remote past whenever the integral exists and has that limit. For example, bounded forcing that tends to zero as

t \to - \infty

satisfies these conditions. A general forcing need not:

f \equiv 1

admits no such solution. A unit-duration pulse with

α = 1

gives

y = 0

before zero,

y = 1 - e^{- t}

between zero and one, and

y = (1 - e^{- 1}) e^{- (t - 1)}

thereafter.

Resonant exponential forcing in a first-order equation ( $lim_{λ \to a} (e^{λ x} - e^{a x}) / (λ - a) = x e^{a x}$ )

For

y' - a y = e^{λ x}

, subtracting an appropriate homogeneous solution from the particular solution removes its apparent singularity as

λ \to a

. The resulting derivative in the parameter produces the resonant particular solution

x e^{a x}

. The arbitrary constant must be reparametrized before taking the limit.

Integrating factor ( $μ (t) = exp (\int_{t_{0}}^{t} a (s) d s)$ )

An integrating factor is a nonzero multiplier that makes a differential equation directly integrable. For a first-order linear differential equation, it gathers the unknown function and its derivative into one product rule expression. Suppose

a

and

b

are continuous functions on an interval containing

t_{0}

, and consider the initial value problem

y^{'} + a (t) y = b (t), y (t_{0}) = y_{0} .

Define

A (t) = \int_{t_{0}}^{t} a (s) d s

and

μ (t) = e^{A (t)}

. The exact solution of a differential equation throughout that interval is

y (t) = e^{- A (t)} [y_{0} + \int_{t_{0}}^{t} e^{A (s)} b (s) d s] .

The definite integral fixes

μ (t_{0}) = 1

, so the initial condition appears without an extra constant to determine. If the equation originally has a coefficient multiplying

y^{'}

, first divide by that coefficient on an interval where it is nonzero; a zero of that coefficient needs separate attention. See OpenStax's first-order linear equations chapter for this normalization and the integrating factor procedure.

For a concrete balance, solve

y^{'} + 2 y = 3

with

y (0) = 0

. Here

μ = e^{2 t}

, so multiplying the entire ordinary differential equation gives

(e^{2 t} y)^{'} = 3 e^{2 t}

. Accumulating from zero to

t

gives

e^{2 t} y (t) = \int_{0}^{t} 3 e^{2 s} d s = \frac{3}{2} (e^{2 t} - 1), y (t) = \frac{3}{2} (1 - e^{- 2 t}) .

The computed values are

y (0.5) \approx 0.94818

y (1) \approx 1.29700

and

y (2) \approx 1.47253

. Its derivative is

3 e^{- 2 t}

, which verifies

y^{'} + 2 y = 3

exactly. Interpreting

3

as a steady input and

2 y

as proportional removal explains the shape: initially all input increases

y

, then removal catches up. The constant particular solution

y_{p} = 3/2

balances input and removal, and the homogeneous solution

y_{h} = - \frac{3}{2} e^{- 2 t}

is the transient response that corrects it to the initial condition. Because this autonomous differential equation has a time-independent right-hand side,

3/2

is also an equilibrium point.

Figure 1.
A steady input becomes a constant particular solution plus a decaying transient
. **Details**.

The first panel adds the particular solution and homogeneous solution; the second shows the same balance in rates,

3 = y^{'} + 2 y

. This links the algebra to a physical accounting rule. Newton's law of cooling has this form with

a = k

and

b = k T_{a}

, while an electrical circuit can have a varying input. In both cases the integrating factor systematically handles the input rather than requiring a guess for a particular solution.

Variable coefficients work just as directly. For

y^{'} + 2 t y = t

y (0) = 0

, the integrating factor is

e^{t^{2}}

, and

(e^{t^{2}} y)^{'} = t e^{t^{2}}, e^{t^{2}} y (t) = \frac{1}{2} (e^{t^{2}} - 1), y (t) = \frac{1}{2} (1 - e^{- t^{2}}) .

(4)

Thus

y (0.5) \approx 0.11060

y (1) \approx 0.31606

and

y (2) \approx 0.49084

. The derivative

y^{'} = t e^{- t^{2}}

verifies

y^{'} + 2 t y = t

, including at zero, where both coefficient and input vanish. No division by

t

is necessary. The general solution is

y = 1/2 + C e^{- t^{2}}

1/2

is a constant particular solution, and

C e^{- t^{2}}

solves the associated homogeneous linear differential equation. Although this special forcing term admits a constant trajectory, this is a nonautonomous differential equation: it depends explicitly on time. Merely setting its right-hand side to zero at one instant would not establish a constant solution of a differential equation; at

t = 0

every state has zero slope.

The direction field shows the selected trajectory leaving its initially horizontal tangent line. For

t \geq 0

, the difference between any two solutions of a differential equation is multiplied by

e^{- t^{2}}

, rather than

e^{- 2 t}

in the constant-rate example. This dependence on elapsed history is clearer if the exact formula is rewritten as

y (t) = y_{0} e^{- \int_{t_{0}}^{t} a (u) d u} + \int_{t_{0}}^{t} e^{- \int_{s}^{t} a (u) d u} b (s) d s .

(5)

For

t \geq t_{0}

, each contribution from the input at

s

is weighted by the accumulated coefficient between

s

and

t

. The homogeneous solution remembers the initial condition; the particular solution given by the second term accumulates the forcing term with zero initial value. This decomposition differs from choosing a constant particular solution, but gives the same total. Initial-data memory tends to zero when

\int_{t_{0}}^{t} a \to + \infty

; a negative accumulated coefficient instead amplifies it. The reciprocal relation between integrating factor and homogeneous solution is developed in MIT's notes on solutions to linear first-order equations.

Why does the trick work? The fundamental theorem of calculus and chain rule give

μ^{'} = a μ

. Consequently the product rule gives

(μ y)^{'} = μ y^{'} + μ^{'} y = μ (y^{'} + a y) = μ b .

(6)

Integrating this identity and dividing by the positive exponential function

μ

produces the stated formula. Any nonzero constant multiple of

μ

works equally well; normalization is a convenience. This is the beauty of the method: a sum involving an unknown function becomes one exact derivative, leaving only an integral of known functions. See Oregon State's product-rule derivation. The related integrating factor for a differential one-form extends the idea of making an expression exact beyond this linear setting.

Limitations of integrating factors for second-order equations ( $w = exp \int (B - A^{'}) / A d x$ )

On an interval where

A \neq = 0

, the displayed factor gives

w (A y^{''} + B y^{'} + C y) = (w A y^{'})^{'} + wC y .

This follows from

(w A)^{'} = wB

. It gives a useful self-adjoint form but does not solve a general second-order ordinary differential equation, since the

wC y

term remains. If

C = 0

, it does reduce to one integration for

y^{'}

. Another reducible class has

C = B^{'} - A^{''}

, because the original left side is then

[A y^{'} + (B - A^{'}) y]^{'}

. These are sufficient special cases, not a classification of all solvable equations.

Heaviside forcing in a first-order integrating-factor equation ( $y (x) = μ (x)^{- 1} [μ (0) y (0) + \int_{0}^{x} μ (t) H (t) d t]$ )

For

y^{'} + a (x) y = H (x)

with continuous

a

, the integrating factor

μ = exp (\int a)

gives the displayed solution. It is continuous at the step and solves the differential equation classically on each side; the derivative jumps by one. The value assigned to the Heaviside step function at the single point zero does not affect the integral. A discontinuity in

y

would instead produce an unintended delta term in its distributional derivative.

Integrating factor for a differential one-form ( $\partial_{x} (μ f) = \partial_{y} (μg)$ )

For the differential one-form

f d y + g d x

, a nonzero integrating factor

μ

makes

μ f d y + μg d x

an exact differential form locally. If it equals

d F

, then

F_{y} = μ f

F_{x} = μg

, and equality of mixed partial derivatives gives the displayed compatibility condition. Conversely this condition is local sufficiency for smooth coefficients on a small rectangle; global exactness additionally depends on the domain. Zeros of the multiplying factor can change the solution set, so equivalence is asserted only where the factor is nonzero.

Integrating factor depending on one coordinate ( $μ^{'} / μ = (P_{y} - Q_{x}) / Q$ )