I've been trying to prove the following as titled:

If I have two linearly independent independent solutions of a homogenous linear system then, $x(t) = v_1(t)x_1(t) + v_2(t)x_2(t)$

$y(t) = v_1(t)y_1(t) + v_2(t)y_2(t)$

will be a particular solution of the inhomogeneous system

$x'(t) = a_1(t)x_1(t) + b_1(t)x_1(t) + f_1(t)$

$y'(t) = a_2(t)y_2(t) + b_2(t)y_2(t) + f_2(t)$

if the functions $v_1$ and $v_2$ satisfy

$v_1'x_1 + v_2'x_2 = f_1$

$v_1'y_1 + v_2'y_2 = f_2$

Now, the proofs I've looked at use the fundamental matrix form. However, I've been struggling hard to find or come up with a proof that doesn't use exponentiated matrices and such. Any help/ideas?

(I read that a proof could be constructed using the same idea as the proofs using the fundamental matrix form without using exponentiated matrices, I'd rather not resort to that just yet <3)



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