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I have the initial value problem $\dot x(t)=Ax(t)+f(t)$ , $x(t_0)=x^0$ , $t \ge t_0$ where $t_0 \in \mathbb{R}$ , $f:[t_0,\infty) \to \mathbb{R}^n$ is piecewise continuous, $A \in \mathbb{R}^{n x n} $ and $x^0 \in \mathbb{R}$.

I know that a unique solution to this inivitial value problem is:

$x(t) = e^{A(t-t_0)}x^0 + \int_{t_0}^t e^{A(t-\tau)}f(\tau) d\tau $

I need to proof why this is a solution. So basically I need to proof the variation of paraters formula.

Any suggestion on how to do that?

user189013
  • 389

1 Answers1

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Look for a particular solution of the form $e^{A(t-t_0)}C(t)$. Substituting in the equation leads to $$ e^{A(t-t_0)}C'(t)=f(t)\implies C'(t)=e^{-A(t-t_0)}f(t). $$

Julián Aguirre
  • 76,354