Solving xy' - y = x tan(y/x) by substitution

[Jim4592]

Homework Statement

Find a general solution if possible, otherwise find a relation that defines the solutions implicity.

xy'-y=x tan(y/x)

Homework Equations

The Attempt at a Solution

y'-(y/x) = tan(y/x)

v = y/x ; y = xv ; y' = xv' + v

xv' + v = tan(v) + v

xv' = tan(v)

I got stuck there I'm not exactly sure how to integrate [tan(v)/x] to get v
and once i get v I'm not sure what to do to get y.

from using mathmatica command DSolve[xy'[x]-y[x]==xTan[y[x]/x],y[x],x] i got an answer of:

y[x] = x*(ArcSin[(e^c)*x])

Any help will be much appreciated!

 

[Dick]

Don't try to integrate tan(v)/x. You've got x*dv/dx=tan(v). Separate the variables. dv/tan(v)=dx/x. Integrate both sides.

 

[Jim4592]

thanks! i forgot all about separating variables! it was so much easier once i did that. i ended up getting the answer, thanks again!