Solve the homogeneous DE $x\,dy/dx = x\, an(y/x) + y$:

Question

Solve the homogeneous DE $x\,dy/dx = x\,	an(y/x) + y$:

Accepted Answer

A. $\sin(y/x) = cx$

Substitute $y = vx$ so $dy/dx = v + x\,dv/dx$. Original: $x(v + x\,dv/dx) = x	an v + vx$ ⇒ $x\,dv/dx = 	an v$ ⇒ $\cot v\,dv = dx/x$. Integrating: $\log\sin v = \log x + \log c$ ⇒ $\sin v = cx$ ⇒ $\sin(y/x) = cx$.

Answer

$\cos(y/x) = cx$

Answer

$	an(y/x) = cx$

Answer

$y/x = cx$

Solve the homogeneous DE $x\,dy/dx = x\,\tan(y/x) + y$: