Solving a linear ODE

1. Aug 31, 2011

NewtonianAlch

1. The problem statement, all variables and given/known data
Solve $\frac{dy}{dx}$ - 2y = x$^{2}$e$^{2x}$

3. The attempt at a solution

Integrating factor = e$^{2x}$

So we multiply through the given equation by the integrating factor and get:

e$^{2x}$$\frac{dy}{dx}$ - 2e$^{2x}$y = x$^{2}$e$^{4x}$

Contract the left-hand side via the chain rule to get:

$\frac{d}{dx}$(e$^{2x}$y) = x$^{2}$e$^{4x}$

Integrate both sides

e$^{2x}$y = $\frac{1}{32}e^{4x}$(8x$^{2}$-4x+1)+C

Now divide through by e$^{2x}$ and the equation definitely does not equal what Wolfram Alpha gives as the solution:

y = $\frac{1}{3}$e$^{2x}$x$^{3}$+Ce$^{2x}$

I checked some of the parts individually with Wolfram, such as the integration of the right-hand side and that was correct, so I'm not too sure what's causing the difference in answers.

2. Aug 31, 2011

NewtonianAlch

Grr, nvm. I forgot the minus sign in the integrating factor -.-