0
$\begingroup$

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?

$\endgroup$

1 Answer 1

1
$\begingroup$

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 $\vec{z}_p(t) + \Phi(t) \vec{c}$.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .