Stumped by Integral: Solving a Differential Equation

[brendan_foo]

Hi guys,

In an attempt to solve the following differential equation, I have come up with an integral that has stumped me.

The differential equation is as follows:

$$\frac{dy}{dx} + \frac{y}{x^2} = 2x$$

Using an integrating factor, I end up with the following:

$$y \cdot e^{-\frac{1}{x}} = 2\int xe^{-\frac{1}{x}} dx$$

I cannot solve that right hand integral, I have tried using parts and substitution and I can't really yield anything meaningful... Is it possible to evaluate this integral using basic calculus methods? Or is something else required?

Thanks!

 

[dextercioby]

Add a constant to this result.

Daniel.

 

[brendan_foo]

I tried using the integrator from Wolfram..but I wanted to see if there was a neater result. This was actually a question on an engineering paper. Have I done steps preceeding the integral evaluation correct, with the integrating factor?

 

[dextercioby]

Everything is okay.Here's what Maple says

$$\frac{dy}{dx}+\frac{y}{x^2}=2x$$

, Exact solution is :

$$y\left( x\right) =x^2-x+e^{\frac{1}{x}}\mbox{Ei}\left( 1,\frac{1}{x}\right) +Ce^{\frac{1}{x}}$$

Daniel.

 

[quasar987]

I get the same thing as you brenden.

Also, I wouldn't trust that integrator too much. It has hapened to me twice that he provided a wrong answer.

 

[brendan_foo]

Ok cheers fellas, must've been a type-o in the paper.

Much appreciated.

 

[brendan_foo]

Oh by the way, I'm not sure what the equivalent is, but I am fairly proficient up to Calc III, and I want to begin pursuing some advance calculus. I am looking for a suitable text in which I can do some self study and tutor myself as best possible. Can anyone recommened a thorough and lucid text for self-study?

I am looking for something more precise, as opposed to just a list of rules and how to implement them.

Thanks guys, much appreciated
Peace