Cannot solve - dy/dx = (e^(1/x))/(x*((x+1)^2))

[manjanjatx]

Homework Statement

Solve the following differential equation:

y' = (e^(1/x)) / (x*((x+1)^2))

Homework Equations

?

The Attempt at a Solution

I wasted over 5 hours trying to solve this equation, but was unable to get a proper solution. Wolfram Alpha and Maple gave me the correct answer (the solution works in the differential equation), but were unable to elaborate. Any help would be appreciated!

 

[Gib Z]

Welcome to PF.

So you just need to integrate your right hand side. From the looks of it, you should first let $u = \frac{1}{x}$ .

 

[dextercioby]

1. Your ODE is

$$\frac{dy}{dx} = \frac{e^{\frac{1}{x}}}{x(x+1)^2}$$

Change the variable $x=\frac{1}{p}$

Separate the new variables and integrate. What do you get ? Post your work.

 

[manjanjatx]

First of all, thank you for the quick replies! I tried a u-substitution again but don't seem to get anywhere:

u = 1/x
dx = - (x^2) du

and v = - (e^u) / (u*((u^-1) + 1)^2) du

And that's as far as I get. Is there something that I am missing?

 

[manjanjatx]

Bump.

Anyone?

 

[dextercioby]

How can you simplify the denominator ?

 

[manjanjatx]

I finally got it; this was one insane integral!

From where I left, the denominator had to be simplified and I had to do another integral (by parts) before I got the answer.

The solution was: y = [(x*exp(1/x)) / (x+1)] + k

Thanks for your help Gib Z and bigubau!