Question

Solve 4y'' - 4y' - 3y = cos(2x) by undetermined coefficients.

Answer 1

So here's the method:
http://wdjoyner.com/teach/DiffyQ/de-undetermined-coeffs.pdf

Answer 2

First: the homogeneous solutions: the characteristic equation is
4r^2 - 4r - 3 = 0
which has roots r = 3/2, -1/2 hence the homogeneous solution is
y = c1.exp(-x/2) + c2.exp(3x/2)

Answer 3

next you need the general form for the guess for yp and that is
yp = A1cos(2x) + A2sin(2x)

(see examples in the link to see why)

Now substitute that into the equation and solve for A1, A2.