dy/dx - 2y = 4x^2

with y(0) = -1

any help would be appreciated

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- Feb 23rd 2010, 05:38 AMbobguydifferential equations
dy/dx - 2y = 4x^2

with y(0) = -1

any help would be appreciated - Feb 23rd 2010, 05:45 AMProve It
This is first order linear, so it is solved using the Integrating Factor.

The integrating factor is

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

Multiply both sides by the integrating factor:

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

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

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

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