How to Solve a Nonlinear ODE using Variable Changes

[ynotidas]

Homework Statement

Solve the nonlinear ODE

du/dx=(u+x√(x^2+u^2 ))/(x-u√(x^2+u^2 ))

by changing variables to x=rcos(theta), u=rsin(theta) and converting the equation to one for d(theta)/dr.

The Attempt at a Solution

Not sure if I'm going in the right direction.

du/dx = du/d(theta) x d(theta)/dx

u = rsin(theta), du/d(theta) = rcos(theta)
x = rcos(theta), dx/d(theta) = -rsin(theta),
i.e. d(theta)/dx = -1/rsin(theta)

so du/dx = rcos(theta)/-1rsin(theta) = - cos(theta)/sin(theta)

then i sub x=rcos(theta), u=rsin(theta) in the main equation

- cos(theta)/sin(theta) = rsin(theta)+rcos(theta)√(r^2sin^2(theta) etc etc..

so i gather the r^2, and make the sin^2+cos^2 both to one, then the √r^2 goes to just 'r'.

then i take divide the whole equation by r

du/dx = -cos(theta)/sin(theta) = [sin(theta)+rcos(theta)]/[rcos(theta)-rsin(theta)

so i times the sin(theta) over the right side and the other to the right..
and i get 1=0...

can someone put me in the right direction on what i did wrong?

thankyou.

 

[ynotidas]

actually the answer is d(theta)/dr = 1, if any of you were wondering.

needed to differentiate x and u in terms of r and theta with the product rule.