After you get a particular solution, don't forget to a generic homogeneous solution (2 unknonw constants!).

You can guess the form of the solution. You did - successfully. Yes - work out what you were trying, and it will give you a solution (I got A = 3/25, B = 4/25). That's easiest and best (when it can be done).

Other approaches:

- There's always the method of variation of parameters. First, you find the homogeneous solution (looks like ae^-x + bxe^-x), then you let the two constants that show up (a and b) become unknown functions (a(x) and b(x)). There's a an integral formula, involving the Wronskian of {e^-x and xe^-x}, for finding that a(x) and b(x).

- You can let y1=y, y2 = y1', and treat it as a system of linear constant coeffs 1st order equations. Then you get something called the fundamental matrix, yadda yadda...

- Series solutions (plug in a generic unknown power series about some point, and you'll get a recursion formula for the terms of that series, solving it in theory (within some radius of convergence anyway).

- Method of Fourier transforms, Laplace Transforms

- The list goes on and on. Plus anything clever you can do that depend on the specifics of the problem.