= Cauchy-Euler differential system
{c}
{title2=$t\mathbf z'=A\mathbf z+t\mathbf b$}
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.
Back to article page
