For a constant matrix $A$, the initial-value problem $x'(t)=Ax(t)$, $x(0)=x_0$, has solution $x(t)=e^{tA}x_0$, with $e^{tA}=\sum_{k\ge0}t^kA^k/k!$. If $A=PDP^{-1}$, then $e^{tA}=Pe^{tD}P^{-1}$. For Jordan blocks, nilpotent terms add polynomial factors in $t$.
Example: $A=\begin{bmatrix}3&1\\1&3\end{bmatrix}$ has eigenpairs $4,(1,1)^T$ and $2,(1,-1)^T$. Hence $x(t)=c_1e^{4t}(1,1)^T+c_2e^{2t}(1,-1)^T$. Eigenvalue real parts control growth or decay.
Higher-order scalar linear ODEs become first-order systems by taking state variables $y,y',\ldots$. Differentiation is linear, but a finite matrix representation requires a specified finite-dimensional setting; the full function space is usually infinite-dimensional.
Parents
Like how this is organised?
Build your own knowledge tree on knowen.md: notes, decisions and tasks in one connected markdown graph, shared with your team and your AI agents.