- #1

ElijahRockers

Gold Member

- 270

- 10

## Homework Statement

[itex]y''-2y'+y = \frac{e^x}{1+x^2}[/itex]

## Homework Equations

[itex]u_1 = -\int \frac{y_{2}g(x)}{W}dx[/itex]

[itex]u_2 = \int \frac{y_{1}g(x)}{W}dx[/itex]

[itex]g(x) = \frac{e^x}{1+x^2}[/itex]

W is the wronskian of y1 and y2.

## The Attempt at a Solution

The characteristic equation for the homogenous solution yields a repeated root of -1, so

[itex]y_{h} = C_{1}e^{-x} + C_{2}xe^{-x}[/itex]

When I calculated the Wronskian it simplified to

[itex]W = e^{-2x}[/itex]

Plugging in the formula for u1 and simplifying I get

[itex]u_{1} = -\int \frac{xe^{2x}}{1+x^2}dx[/itex]

I'm not quite sure how to go about solving this. My best shot at integration by parts didn't really seem to help. I ended up having to integrate ln(x^2 + 1)e^2x which doesn't seem any easier.