y''+y' -2y=2-2e^-2x ...... find general soln
roots = 1 & -2
Yc = c1 e ^x + c2 e^-2x
"Variation of parameters" always works but is typically much harder than "undetermined coefficients" when that works- which it will as long as the right hand side consists of the kind of functions we get for homogeneous linear equations with constant coefficients, polynomials, exponentials, sine and cosine, or products of those.
Here, your right hand side is . Because of the constant "2", try some constant A. Because of the " ", your first thought should be " ". However, is already a solution to the associated homogeneous equation so try " " instead.
In other words, try .
If , then
Even with " " this would not be the second derivative. You are aware that the derivative of -2A is 0, aren't you?yp''= 4(A+Bxe^-2x)
You should have
therefore from orginal equation
4(A+Bxe^-2x) + -2(A+Bxe^-2x) - 2*(A+Bxe^-2x) = 2-2e-2x