Math Help Forum - View Single Post

Jhevon · #21 July 11th, 2007, 09:00 PM

Quote:

Originally Posted by emily28

does anyone have any suggestions/

I will say what Krizalid said, but in a more general way.

For a differential equation of the form: $y' + p(x)y = f(x)$

we define $\mu (x) = e^{\int p(x)~dx} = \mbox { exp} \left( \int p(x)~dx \right)$ as the integrating factor of the differential equation. We solve the differential equation by multiplying through by the integrating factor. so we get the new equation:

$\mu (x) y' + \mu (x) p(x) y = \mu (x) f(x)$

it will always be the case that the left hand side is the derivative we would get from applying the product rule. so the equation becomes:

$\left( \mu (x) y \right)' = \mu (x) f(x)$

$\Rightarrow \mu (x) y = \int \mu (x) f(x)~dx$

$\Rightarrow y = \frac { \int \mu (x) f(x) ~dx}{ \mu (x)}$ is our solution

remember that you will get an arbitrary constant from the integration, you have to divide this by $\mu (x)$ as well