SOLVED Integration by Parts Question

[Monocles]

[SOLVED] integration by parts question

Homework Statement

I think my first problem is at integration by parts, but let me know if you see a different error in my work.

Edit: I shoud mention that this is Calculus 2 and we just learned first order differential equations today. So I don't have any fancy methods available.

Find the solution of
$$x^3 y'+ax^2 y = 2x^{-a+4} \sin(x)$$
with
$$y(\pi) = 0$$

Homework Equations

Format of first order differential equations:
$$y' + P(x) y = q(x)$$

$$H(x) = \int P(x) dx$$

The Attempt at a Solution

$$x^3 y'+ax^2 y = 2x^{-a+4} \sin(x)$$

$$y' + \frac{a y}{x} = \frac{2 x^{-a + 4} \sin(x)}{x^3}$$

$$y' + \frac{a y}{x} = 2 x^{-a + 1} \sin(x)$$

$$H(x) = \int \frac{a}{x} dx$$

$$H(x) = a \ln{x}$$

Take $$e^{H(x)} = x^a$$ and multiply it to both sides

$$x^a y' + \frac{a y}{x} x^a = x^a 2 x^{-a + 1} \sin(x)$$

$$(x^a y)' = 2 x^{-a^2 + a} \sin(x)$$

$$x^a y = \int 2 x^{-a^2 + a} \sin(x) dx$$

This is where I am stuck. The only way I can think of to solve that integral is integration by parts, but the only way I can see that working is by repeating it enough times for the part of the integral before sin(x) to disappear. So I think I went wrong somewhere? But I don't see where.

 

[Rainbow Child]

$$x^a y' + \frac{a y}{x} x^a = x^a 2 x^{-a + 1} \sin(x)$$

leads to

$$(x^\alpha\,y)'=2\,x\,\sin x$$

 

[Monocles]

Ohh... Well that was easy. I just forgot how to multiply! Thanks.