Help! Solving a Differential Equation with Laplace

[silvrstring]

Hello everyone.

I think Pierre-Simon Laplace is alive and well and he is trying to kill me, or drive me mad. I am trying to solve the attached problem. It is a differential equation---D2x+Dx-2*x=5*e^(-t)*sin(t). I have repeatedly tried to solve this problem. I don't like asking for help with homework, but I desperately want to know what I am doing wrong. I would like to think that the book has the wrong answer (it has happened before). Unfortunately, I don't have access to my MATLab, right now---I don't know why. So, I can't check it.

The problem, and one of my several attempts at the solution, are attached. I hope you can help me see my error(s).

 

[CompuChip]

Unfortunately I can't see the attachment yet. But generally one would first solve the homogeneous problem and try to find a particular solution.
The homogeneous problem is
$$x''(t) + x'(t) - 2 x(t) = 0$$
so the standard approach is plugging in $x = e^{\lambda t}$ as a trial solution.
For the inhomogeneous problem, you have something with trig functions, so I'd try plugging in $x = \left[ A cos(t) + B sin(t) \right] e^{-t}$ and try to solve for A and B to find a particular solution.

Otherwise, I'm waiting for the attachment to be approved :)

 

[Defennder]

I got the same answer you did. Seems like there's something wrong with the book's answer. By the way, being lazy, I used the online calculators at wims:
http://wims.unice.fr/wims/

 

[silvrstring]

Welp, the book was wrong. The written answer is correct. Thanks for responding, though.
I'll check out the wims calculator Defennder. Thanks for the tip.