allerySolve the homogeneous DE $x\,dy/dx = x\,\tan(y/x) + y$:
A$\sin(y/x) = cx$
B$\cos(y/x) = cx$
C$\tan(y/x) = cx$
D$y/x = cx$
Answer & Solution
Correct answer: A. $\sin(y/x) = cx$
Substitute $y = vx$ so $dy/dx = v + x\,dv/dx$.
**Original:** $x(v + x\,dv/dx) = x\tan v + vx$
⇒ $x\,dv/dx = \tan v$
⇒ $\cot v\,dv = dx/x$.
**Integrating:** $\log\sin v = \log x + \log c$
⇒ $\sin v = cx$ ⇒ $\sin(y/x) = cx$.
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