Second Order, Non Homogeneous Linear Differential Equation - Particular Integral

I have been presented with a problem, as described above, with a difficult right hand side of e^{x}cosx; an example would be y'' + 2y' +y = (e^{x}cosx). I have utilized the auxiliary equation to find the complementary solution, but which form do I need to use to find my particular integral? I have begun to use Ae^{x(}cosx) + Be^{x(}sinx), but I'm fairly certain that isn't right. Any help would be fantastic!

Re: Second Order, Non Homogeneous Linear Differential Equation - Particular Integral

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.

Re: Second Order, Non Homogeneous Linear Differential Equation - Particular Integral

Thanks so much dude, I had a feeling it would be what I put down but I wasn't sure. Much appreciated!

Re: Second Order, Non Homogeneous Linear Differential Equation - Particular Integral

$\displaystyle y(x)= c_1 e^{-x}+c_2 x e^{-x} +\frac{1}{25} e^x [4 \sin (x)+3 \cos (x)]$