How Do You Prove a General Solution for Non-Homogeneous Differential Equations?

[lalaman]

Hello all, I am currently having trouble with this Differential Equations problem.

Let x = F(t) be the general solution of x'=P(t)x+g(t), and let x=V(t) be some particular solution of the same system. By considering the difference F(t)−V(t), show that F(t)=U(t)+V(t), where U(t) is the general solution of the homogeneous system x'=P(t)x.

Attempt:
Since F is a solution, we know that there exists a fundamental matrix such that M such that F=MW, where W is such that MW′=g. But that is all I have been able to deduce. Also, I am not sure if F(t) - V(t) would be considered a soluton as well.

Thank you for your time. :)

 

[Simon Bridge]

Welcome to PF;
It looks like you are being asked to prove the usual theorem that is used to solve non-homogeneous DEs - generalized for a system of DEs. You can always look up how it is normally done for clues.

Label your equations - (1) is the inhomogeneous equation and (2) is the associated homogeneous one.
So F is the general solution to (1) and V is a particular solution to (1).
You can easily check to see if F-V is a solution to (1) - plug it in.