Linear ordinary differential equation

A linear ordinary differential equation is linear in the unknown function and its derivatives.

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.

Suppose two complex amplitudes satisfy ${\partial }_t|A{|}^2\le 2a|A||B|-2d|A{|}^2$ and ${\partial }_t|B{|}^2\le 2c|A||B|-2d|B{|}^2$, with positive $a,c$ and $d\ge 0$. Then $N=c|A{|}^2+a|B{|}^2\ge 2\sqrt{ac}|A||B|$ gives $\dot{N}\le 2(\sqrt{ac}-d)N$. 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 models the temperature $T$ of an object in an environment at constant temperature $T_{\infty }$ by $\dot{T}=-a(T-T_{\infty })$ for a positive cooling rate $a$.

In an ideal linear temperature-dependent destruction model with equal warming and cooling durations, normalized temperature is $1-e^{-\alpha t}$ during warming and $(1-e^{-\alpha T})e^{-\alpha (t-T)}$ 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.

Under Newton's law of cooling with fixed ambient temperature $T_0$ and $k>0$, an instantaneous jump $\beta$ at $t_{\ast }$ sets $T(t_{\ast }+)=T(t_{\ast }-)+\beta$. The subsequent temperature is $T_0+[T(t_{\ast }-)+\beta -T_0]e^{-k(t-t_{\ast })}$. Equivalently, for an initial temperature $T_i$, the full solution isThis piecewise model has the distributional derivative equation $T^{\prime }+k(T-T_0)=\beta \delta (t-t_{\ast })$, with the Dirac delta distribution representing the added temperature.

A heat transfer coefficient relates heat flux through a boundary to a temperature difference: $F=\lambda \Delta T$. Its dimensions are power per area per temperature. In a quasistatic ice layer, atmospheric boundary resistance $1/{\lambda }_A$ and conductive resistance $h/k$ add in series, giving effective conductance $(1/{\lambda }_A+h/k{)}^{-1}$.

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 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.

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.

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.

A second-order linear equation has the form $y\prime \prime +p(x)y\prime +q(x)y=f(x)$.

The homogeneous equation has linearly independent solutions $e^{rx}$ and $x{e}^{rx}$, with nonzero Wronskian $e^{2rx}$. Setting $y=e^{rx}u$ converts the forced equation to $u^{\prime \prime }=e^{-rx}f$. For resonant forcing $f=e^{rx}$, the particular response is $x^2{e}^{rx}/2$. This substitution explains the extra powers of $x$ required by repeated-root resonance.

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.

For $t>0$, the change of variables $\tau =1/t$ and $u(t)=v(\tau )/\tau$ converts $t^4{u}^{\prime \prime }+{\lambda }^{2}u=0$ into $v^{\prime \prime }+{\lambda }^{2}v=0$. The chain rule gives $u^{\prime }=v-\tau v^{\prime }$ and $u^{\prime \prime }={\tau }^3{v}^{\prime \prime }$. Thus $u=t[A\cos (\lambda /t)+B\sin (\lambda /t)]$. The prefactor cancels the derivative term introduced by inverting the independent variable.

A smooth invertible change of variables in $u_{xx}-f(x)u=0$ introduces a first derivative in time. Rescaling by ${\dot{x}}^{-1/2}$ removes it. Choosing $\dot{x}=f^{-1/2}$ 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.

For $\gamma \ne 0$ and $x>0$, the change of variables $u=x^{\gamma }$ convertsinto $Y_{uu}+Y=0$, where $y(x)=Y(u)$. The chain rule cancels the first-derivative terms. Thus $\cos (x^{\gamma })$ and $\sin (x^{\gamma })$ form a pair of linearly independent solutions, with Wronskian $\gamma x^{\gamma -1}$. This is a variable-speed reparametrization of the harmonic oscillator equation.

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 $t$ by $t/2$ rescales both the constant and periodic coefficients.

The Mathieu characteristic values are the parameter values $a$ for which the Mathieu equation has the even periodic mode traditionally denoted ${\mathrm{ce}}_n$, or the odd periodic mode ${\mathrm{se}}_n$. They are eigenvalues of $-d^2/d{t}^2+2q\cos 2t$ 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.

This second-order ordinary differential equation has the independent entire Airy functions $\mathrm{Ai}$ and $\mathrm{Bi}$ as a standard basis. It is distinct from the dispersive PDE in Airy equation. Its general solution is $a_1\mathrm{Ai}(Z)+a_2\mathrm{Bi}(Z)$.

For the Airy ordinary differential equation $y^{\prime \prime }=xy$, a power series obeys $a_2=0$ and $a_{m+3}=a_m/[(m+3)(m+2)]$. Setting its first two coefficients to $(1,0)$ or $(0,1)$ givesEmpty 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.

The standard Airy functions solve the Airy ordinary differential equation. Their Wronskian is $\mathrm{Ai}(Z){\mathrm{Bi}}^{\prime }(Z)-{\mathrm{Ai}}^{\prime }(Z)\mathrm{Bi}(Z)=1/\pi$, so they form a fundamental solution pair, including for complex $Z$.

For $q=e^{i\pi /6}$ define $I_0(C)={\int }_0^{\infty }\mathrm{Ai}(q(s-C))ds$ and $I_1(C)={\int }_0^{\infty }s\mathrm{Ai}(q(s-C))ds$. Both converge by Airy function decay. Integrating its differential equation gives $I_1-C{I}_0=iq\mathrm{Ai}\prime (-qC)$. These moments convert a decaying vorticity profile into its velocity and displacement for three-layer long-wave shear-flow matching.

For a second-order linear differential equationthe substitution $y=u\exp [-\frac{1}{2}\int p(x)dx]$ eliminates the first-derivative term and gives

If $q_1\le q_2$, then between consecutive zeros of a nontrivial solution ofthere is a zero of every nontrivial solution of $y^{\prime \prime }+q_2(x)y=0$, unless the two coefficients agree throughout that interval.

For solutions ${\phi }_i^{\prime \prime }+q_i{\phi }_i=0$, the mixed WronskiansatisfiesIts endpoint signs between consecutive zeros of a positive ${\phi }_1$ force a zero of ${\phi }_2$ unless $q_1=q_2$.

Kummer's equation isIts solution analytic at zero is the confluent hypergeometric function of the first kindWhen $b$ is not an integer, a second local solution is

Confluent hypergeometric functions solve the Kummer differential equation $z{y}^{\prime \prime }+(b-z)y^{\prime }-ay=0$. 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.

For $b$ not a nonpositive integer, the regular solution normalized by $M(a,b,0)=1$ has the power seriesHere $(a{)}_n$ is the rising factorial. Substitution into the Kummer differential equation proves the coefficient recurrence. This normalization agrees with NIST's definitions.

For $y\prime \prime +py\prime +qy=0$, the Wronskian satisfies $W\prime =-pW$.

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

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.

The integral representation $y_2=y_1\int e^{-\int p}/y_1^2$ is derived where $y_1\ne 0$. 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 $y_2=x\mathrm{arctanh}x-1$ illustrate this cancellation at zero.

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.

Forthe power ansatz $w=z^{\lambda }$ gives the indicial equation $\lambda (\lambda -1)+a\lambda +b=0$. Distinct roots give $C_1{z}^{{\lambda }_1}+C_2{z}^{{\lambda }_2}$; a repeated root $\lambda$ gives $z^{\lambda }(C_1+C_2\log z)$ on a chosen logarithm branch.

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 $s$.

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

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.

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.

A log-periodic oscillation is periodic in the logarithm of its argument, for example $\cos (\mu \log x)$. Its successive zeros or extrema therefore occur at radii in a geometric progression.

The equation $(1-x^2)y\prime \prime -2xy\prime +\ell (\ell +1)y=0$ has polynomial solutions for nonnegative integer $\ell$.

Associated Legendre functions solve $(1-x^2)y^{\prime \prime }-2x{y}^{\prime }+[\nu (\nu +1)-{\mu }^2/(1-x^2)]y=0$. For integers $\ell \ge 0$ and $0\le m\le \ell$, the regular solution is $P_{\ell }^m(x)=(-1{)}^m(1-x^2{)}^{m/2}(d/dx{)}^m{P}_{\ell }(x)$, 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 $P_{\ell }^{-m}=(-1{)}^m(\ell -m)!P_{\ell }^m/(\ell +m)!$.

The polynomial solution of degree l normalized by P_l(1)=1 is the Legendre polynomial P_l.

The Legendre polynomials are orthogonal on $[-1,1]$ with unit weight:

The three-term recurrenceturns multiplication by $x$ into coupling between adjacent Legendre multipoles.

For a contour enclosing $t$,

For fixed $0<\theta <\pi$,

For fixed $\xi >0$,

For $z=z(x)$, the second derivative transforms as $w_{xx}=z_x^2{w}_{zz}+z_{xx}{w}_z$.

A first-order linear differential equation has the form $y^{\prime }+p(x)y=q(x)$. Multiplication by an integrating factor $\mu$ satisfying ${\mu }^{\prime }=p\mu$ turns it into $(\mu y{)}^{\prime }=\mu q$.

For $y\prime -ay=e^{\lambda x}$, subtracting an appropriate homogeneous solution from the particular solution removes its apparent singularity as $\lambda \to a$. The resulting derivative in the parameter produces the resonant particular solution $x{e}^{ax}$. The arbitrary constant must be reparametrized before taking the limit.

An integrating factor turns a first-order linear equation into an exact derivative.

For a first-order linear equation whose homogeneous solution grows at infinity, boundedness fixes the integration constant by rewriting the particular integral as a tail integral from the current point to infinity.

Resonance occurs when forcing overlaps a homogeneous mode, requiring multiplication of the usual particular ansatz by an extra power or logarithm.

A resonance condition is the relation between forcing parameters and a homogeneous mode that makes the forcing operator singular on that mode. A repeated inversion then produces algebraic amplitude growth.

When forcing matches a natural frequency, the oscillator response acquires a linearly growing amplitude.

A linear differential system has vector form $x\prime =A(t)x+b(t)$.

An eigenvector of $A$ with eigenvalue $\lambda$ gives a homogeneous solution $t^{\lambda }\mathbf{v}$ for $t>0$. If $A$ is diagonalizable, these powers form a fundamental system. A constant vector forcing has a particular solution $t(I-A{)}^{-1}\mathbf{b}$ when $1$ is not an eigenvalue. At a resonant eigenvalue or a nontrivial Jordan block, logarithmic factors in $t$ can occur, equivalently by changing time to $\log t$ and solving a constant-coefficient linear system.

For a constant coefficient system, eigenvectors split the homogeneous equation into scalar exponential or power-law modes.

A state transition matrix maps the state of a homogeneous linear system between two times.

Encoding two planar components as one complex variable turns a rotation-dilation system into a scalar complex equation.

Integrating an equation through a Dirac impulse determines the jump in the derivative or state while nonsingular lower-order terms contribute no jump.

For the harmonic oscillator equation $y^{\prime \prime }+{\omega }^{2}y=J\delta (t-t_{\ast })$, with $\omega >0$, the displacement is continuous and the velocity increases by $J$ at the impulse. If the incoming displacement is zero and $J=-y^{\prime }(t_{\ast }-)$, the outgoing displacement and velocity both vanish. The subsequent homogeneous solution is identically zero. Equivalently, the causal forced response $(J/\omega )H(t-t_{\ast })\sin (\omega (t-t_{\ast }))$ cancels the pre-existing oscillation, where $H$ is the Heaviside step function.