Duhamel formula for a linear ODE

For continuous $A,f$ , the solution of $y'=A(t)y+f(t)$ , $y(s)=y_s$ , is

y(t)=\Phi(t,s)y_s+\int_s^t\Phi(t,\tau)f(\tau)\,d\tau,

where $\Phi$ is the principal propagator of the homogeneous system. This is Duhamel's formula, also called variation of constants.

Verification

Differentiate the integral: its moving endpoint contributes $f(t)$ , and differentiating the propagator contributes $A(t)$ times the integral. The initial condition follows at $t=s$ . Equivalently, differentiating $X(t)^{-1}y(t)$ gives $X(t)^{-1}f(t)$ . Bounds on $\Phi$ therefore turn source bounds into solution bounds.