How to show for a linear ODE system, the solution can be given as the linear relation $p(t)=K(t)x(t)+s(t)$??

Question

Suppose we have the following ode system: $$\begin{bmatrix} \dot x(t) \\ \dot p(t) \\ \end{bmatrix} = \begin{bmatrix} a_{11} && a_{12} \\ a_{21} && a_{22} \\ \end{bmatrix}\begin{bmatrix} x(t) \\ p(t) \\ \end{bmatrix} +\begin{bmatrix} r_1(t) \\ r_2(t) \\ \end{bmatrix};$$ is it always correct to suppose $p(t)=K(t)x(t)+s(t)$? How can we show this relation holds?

Answer 1

Assuming $p(t) = Kx(t)+s(t)$. If your coefficients are constant, then: $$\frac{dp}{dt} = K\frac{dx}{dt} + s'(t) = a_{21}x+a_{22}p + r(t)$$ implies that: $$Ka_{11} = a_{21}, Ka_{12} = a_{22}.$$ If your coefficient matrix is not constant, you can work the same.