Solving First Order Linear ODE: dy/dx = y/x + tan(y/x)

[whatisreality]

Homework Statement

Solve dy/dx = y/x + tan(y/x)

Homework Equations

The Attempt at a Solution

Not separable, as far as I can tell. It's not homogeneous, since for the tan term f(λx,λy) = tan(λy/λx) = tan(y/x) ≠ λtan(y/x). It's also not of the form dy/dx + P(x)y = Q(x), because I don't think Q(x) should involve y. And that completes the list of methods I know, none of which I can use! How do you solve this?! Is there a substitution I should make?

 

[HallsofIvy]

Even though this is not homogeneous, seeing that "y/x" my first thought would be to try the substitution u= y/x.

Then y= xu so that dy/dx= u+ x du/dx. The differential equation becomes u+ x du/dx= u+ tan(u) so that x du/dx= tan(u).

That is separable.

 

[stevendaryl]

You could try making a change in the dependent variable, from $y$ to $u$, where $u= y/x$

 

[whatisreality]

OK, that substitution works! I thought it was only for homogeneous equations. Thanks :)