# Ordinary Differential Equations

• NeoDevin

## Homework Statement

Use variation of parameters to find the general solution of:

$$x^2 y'' + xy' + 9y = -tan(3ln(x))$$

## Homework Equations

$$y_p = y_2 (x) \int \frac{y_1 (x) g(x)}{W[y_1 (x),y_2 (x)]} dx - y_1 (x) \int \frac{y_2 (x) g(x)}{W[y_1 (x),y_2 (x)]} dx$$

## The Attempt at a Solution

Solution for the homogeneous Euler equation is given by
$$y_1 (x) = cos(3ln(x)), y_2 (x) = sin(3ln(x))$$

I try using these solutions in the equation for variation of parameters with:
$$W[y_1 , y_2] = \frac{3}{x}$$
$$g(x) = -tan(3ln(x))$$

The first term I can evaluate, but the second term gives me:
$$y_1 (x) \int \frac{x sin^2 (3ln(x))}{3 cos(3ln(x))} dx$$

which doesn't seem solvable by any methods I know...

Any suggestions on what to do from here, or where I went wrong on the way would be helpful.

You have to set it up as y" +... to use your variation of parameters formula

so the solutions for the homogeneous equation are the same, right? but I need to divide through by $x^2$ before I apply the formula? so
g(x) becomes:

$$g(x) = -\frac{tan(3ln(x))}{x^2}$$