# Math Help - differential equations

1. ## differential equations

dy/dx - 2y = 4x^2

with y(0) = -1

any help would be appreciated

2. Originally Posted by bobguy
dy/dx - 2y = 4x^2

with y(0) = -1

any help would be appreciated
This is first order linear, so it is solved using the Integrating Factor.

The integrating factor is

$e^{\int{-2\,dx}} = e^{-2x}$.

Multiply both sides by the integrating factor:

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

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

$e^{-2x}y = \int{4x^2e^{-2x}\,dx}$

Now evaluate the right hand side using integration by parts (twice).