Solve differential equation $y'-y/x= \tan(y/x)$ using integrating factor

Question

Solve differential equation using integrating factor $$y'-y/x= \tan(y/x)$$

I get the integrating factor to be $1/x$

Multiply that out and the LHS $= d/dx(y (1/x))$ and the RHS then becomes $\tan(y/x) (1/x)$

Don't know where to go from that.

Answer 1

Don't agree with 1/x as integrating factor. Following Chappers idea works perfect.

$\frac{\text{dy}}{\text{dx}}=\frac{y}{x}+\tan \left(\frac{y}{x}\right)$

Substitude:

$\frac{y}{x}=u$

$y=x u$

$\frac{\text{dy}}{\text{dx}}=x\frac{\text{du} }{\text{dx}}+u$

$ x\frac{\text{du}}{\text{dx}}+u=u+\tan (u)$

$ x\frac{\text{du}}{\text{dx}}=\tan (u)$

$\frac{\text{du}}{\tan (u)}=\frac{\text{dx}}{x}$

Now on both sides logarithmic derivatives:

$cos (u)\frac{\text{du} }{\sin (u)}=\frac{\text{dx}}{x}$

$\frac{d}{\text{du}}(\ln \sin (u))=\frac{d}{\text{dx}}(\ln (x))$

$\ln (\sin (u))=c+\ln (x)$

Solving y should be easy: $\sin (u)=C x$

$u=\arcsin (C x)$

$y=x \arcsin (C x)$

In my opinion, conditions for integrating factor are hard to solve.