Variation of parameters

Consider

$$\dot X(t)=AX(t)+G(t), X(t_0)=x_0,$$

where A is an $n\times n$constant matrix. We have a fundamental matrix for $\dot X=AX$: e^At. We substitute $X(t)=\exp(At)u(t)$, and obtain

(e^Atu(t))'=Ae^Atu(t)+G(t),

which yields e^Atu'(t)=G(t) so that u'=e^-AtG(t). This is integrated to obtain

$$u(t)-u(t_0)=\int_{t_0}^t{e^{-As}G(s) ds} .$$

Hence (note that u(t₀)=x₀),

$$X(t)= e^{At}u=e^{At}x_0 + e^{At}\int_{t_0}^t{e^{-As}G(s) ds}$$

The first term is a complementary solution. The last term is the particular solution and can be written

$$\int_{t_0}^t{e^{A(t-s)}G(s) ds}.$$

This is a convolution of two functions, e^At and G(t).