In this PatrickJMT video, he doesn't seem to use the Wronskian. He produces a set of equations and solves the example with them, but I'm not exactly sure where the equations come from.
Everywhere else I look seems to solve these kinds of problems using a Wronskian (and I think I have to, too, as I don't have the fundamental set of solutions given to me in the question like Patrick does).
Could anyone explain to me where the PatrickJMT method comes from?
Of course he used the Wronskian: $$c'_1y_1+c'_2y_2=0$$ $$c'_1y'_1+c'_2y'_2=f$$ With matrix notation we have: $$\pmatrix { y_1 & y_2 \\ y'_1 &y'_2}\pmatrix {c'_1 \\c'_2}=\pmatrix {0 \\ f}$$ $$\pmatrix {c'_1 \\c'_2}=\pmatrix { y_1 & y_2 \\ y'_1 &y'_2}^{-1}\pmatrix {0 \\ f}$$ $$\pmatrix {c'_1 \\c'_2}=\dfrac 1 W\pmatrix { y'_2 & -y_2 \\ -y'_1 &y_1}\pmatrix {0 \\ f}$$
Where $W$ is the Wronskian.