Linear system of differential equations

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.