Suppose I have a fundamental matrix $\Phi(t)$ of the homogeneous system $y'=A(t)y$. I am trying to show the derivation for the variation of constants method of solving non-homogeneous systems $y'=A(t)y+g(t)$.
Suppose $\psi(t)=\Phi(t)v(t)$ is a solution to the non-homogeneous equation for some non-constant vector $v(t)$.
Then $\Psi'(t)=\Phi(t)v(t)+\Psi(t)v'(t)=A(t)\Phi(t)v(t)+g(t)$
My textbook continues to say "Since $\Phi$is a fundamental matrix of the homogeneous system, $\Phi'(t)=A(t)\Phi(t)$..."
My question is, is this claim a fundamental property of a fundamental matrix, as the text seems to suggest, or is it an argument made by matching the terms attached to $v(t)$? i.e. $\Phi(t)v(t)=A(t)\Phi(t)v(t)$?
If $\Phi$ is a fundamental matrix of the system $y'=A(t)y$, the each column vector of $\Phi$ is a solution. We can write $\Phi$=[$\phi_1,\phi_2$], where $\phi_1$ and $\phi_2$ solve the homogeneous system.
I.e. $\phi'_1=A(t)\phi_1$, and $\phi'_2=A(t)\phi_2$.
Therefore $\Phi'$=[$\phi'_1,\phi'_2$]=[$A(t)\phi_1,A(t)\phi_2$]=$A(t)$[$\phi_1,\phi_2$]=$A(t)\Phi$...
I swear the hardest thing about differential equations is the notation.