Linear differential equation

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

For a system of linear differential equations $u^{\prime }=A(t)u$, a fundamental matrix has columns forming a basis of its solution space. It solves $X^{\prime }=A(t)X$ and is invertible at every time in its interval. Every solution is $u(t)=X(t)c$ for a constant vector $c$: differentiating $X^{-1}u$ gives zero. Invertibility at one time implies invertibility throughout the interval by uniqueness of the initial value problem. This notion is distinct from a Markov-chain fundamental matrix or the projective-geometric matrix used in stereo vision.

If $\dot{\Phi }=A(t)\Phi$ and $\Phi (0)=I$, differentiation of the determinant gives $\det \Phi (t)=\exp ({\int }_0^t\mathrm{tr}A(s)ds)$. For a planar periodic orbit this gives the nontrivial Floquet multiplier, since the tangent multiplier is one.

For a continuous periodic coefficient matrix $A(t+L)=A(t)$, a fundamental matrix $X$ for $X^{\prime }=A(t)X$, normalized by $X(0)=I$, obeys $X(t+L)=X(t)X(L)$. The invertible matrix $X(L)$ has a complex matrix logarithm; set $B=L^{-1}\log X(L)$ and $P(t)=X(t)e^{-tB}$. Since $X(L)=e^{LB}$ commutes with $B$, direct substitution gives $P(t+L)=P(t)$ and proves the factorization. A real logarithm is not required for this complex representation.

The eigenvalues of $X(L)$ are Floquet multipliers. All solutions are bounded for forward time when every multiplier has modulus at most one and those on the unit circle have no nontrivial Jordan blocks. In a real undamped second-order scalar equation with no first-derivative term, the monodromy determinant is one. Distinct conjugate unit-modulus multipliers give bounded solutions; a nontrivial Jordan block at multiplier $1$ or $-1$ gives an unbounded second solution even when one periodic or antiperiodic solution exists.

The maximal asymptotic growth rate of a finite-dimensional periodic linear differential equation is $P^{-1}\log \rho (N)$, where $P$ is the coefficient period and $N$ the monodromy matrix of a periodic linear system. The spectral radius selects the dominant Floquet multiplier. Generic initial data attain this rate; data lying entirely in other invariant subspaces can grow or decay at different rates.

For $\dot{Y}=K(t)Y$ with coefficient period $P$, the monodromy matrix maps $Y(0)$ to $Y(P)$. Its eigenvalues are the Floquet multipliers. For piecewise constant coefficients it is the time-ordered product of the segment matrix exponentials, with the earliest segment on the right. Its determinant is $\exp ({\int }_0^P\mathrm{tr}K(t)dt)$.

For the homogeneous linear differential equation $P({\partial }_x)u=0$, a characteristic root is a root of a polynomial $P(r)=0$. A simple root supplies an exponential solution $e^{rx}$; a root of multiplicity $m$ supplies $x^j{e}^{rx}$ for $0\le j<m$. The sign of the real part of $r$ determines spatial growth or decay.

The Laplace contour-integral method seeks a solution in the formReplacing multiplication by $z$ with differentiation of $e^{zt}$ in $t$ and integrating by parts turns the differential equation for $w$ into a first-order equation for $f$. The contour $\gamma$ must make the resulting endpoint term vanish; finite endpoints are often branch points of $f$, while infinite ends must enter sectors where the exponential decays.

A homogeneous linear differential equation has zero forcing term. Its solutions are closed under addition and scalar multiplication.

The solutions of a homogeneous linear differential equation on a common domain form a vector space. For a regular scalar equation of order $n$, prescribing the first $n$ initial data at an ordinary point gives an isomorphism of this solution space with the scalar field to the power $n$.

A linearized equation retains terms first order in a perturbation about a reference solution. Its coefficients are determined by derivatives of the original nonlinear equation evaluated on that reference solution.

An inhomogeneous linear differential equation has a nonzero forcing term. Its general solution is one particular solution plus the general solution of the associated homogeneous equation.