The method of undetermined coefficients (I)

Problem:

$y'' - y = \cos x$

Solution:

Let's first solve $z'' - z  = 0$, using the characteristic polynomial method, we know the answer is $z = Ae^x + Be^{-x}$.

It remains to find just one function that fits $w'' - w = \cos x$, then we know $y = w + z$ and have the degree of freedom we need.

At this point, the so called "method of undetermined coefficients" is really just about guessing the solution, let's guess our solution is $w = \cos x$ and see

$(\cos x)'' - \cos x = -2 \cos x$.

So now the solution is obviously $w = \frac{-1}{2}\cos x$

The full solution is then $Ae^x + Be^{-x} - \frac{1}{2}\cos x$.