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I am currently taking a class on Fundamentals of Differential Equations. The textbook is of the same name and authored by Nagle, Saff, and Snider.

I don't completely understand the derivation of the variation of parameters method for solving an ODE. Particularly the part where we assume that the particular solution of the ODE is of the form $v_1(t)y_1(t) + v_2(t)y_2(t)$.

What allows this assumption? In my opinion it seems like a huge assumption to make.

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Nothing technically allows it, you're just asking what conditions would $v_1$ and $v_2$ have to satisfy if they happened to exist, and you get to a pair of first order differential equations. Now, if said pair of ODEs happened to have a solution, you would have a solution for you inhomogeneous problem. The theorem of existence and uniqueness for first order ODEs guarantees there is a pair of solutions.

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