# System of ODE - fundamental matrix

I'm not sure how a definite integral is used in this context. Say I have the system $$\vec{z}^{'}(t) = A\vec{z}(t) +\vec{a}(t)$$ where A is a matrix. Then the inhomogeneous solution is generally $$\vec{z_p}=\Phi(t)\int \Phi^{-1}(t)\vec{a}(t)dt$$ where $$\Phi$$ is the fundamental matrix. I can find the integration constant using given initial values. But I sometimes see it written like this: $$\vec{z_p}=\Phi(t)\int_{t_{0}}^t \Phi^{-1}(m)\vec{a}(m)dm$$ Where m is some dummy variable and $$t_0$$ is that of the initial condition. How does that generally get us the same result as the indefinite integral?

Analogous to the usual calculus result, for a vector-valued function the definite integral $$\int_{t_0}^t \vec{F}(m) \; dm$$ is one antiderivative of $$\vec{F}(t)$$, and the general antiderivative is $$\int_{t_0}^t \vec{F}(m)\; dm + \vec{c}$$ where $$\vec{c}$$ is an arbitrary vector. So your $$\vec{z}_p$$ is one particular solution of the differential equation (that subscript $$p$$ stands for "particular"), and the general solution is $$\vec{z}_p(t) + \Phi(t) \vec{c}$$.