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Solve dy/dx = y/x + tan(y/x)
Short answer
Homogeneous. Put y=vx to get cot v dv = dx/x. Integrate: ln|sin v| = ln|x|+ln c, so sin v = cx. Solution: sin(y/x) = cx.
Solve the differential equation dy/dx = y/x + tan(y/x).
The right side is a function of y/x, so the equation is homogeneous.
Substitution
Put y = vx, so dy/dx = v + x dv/dx. The equation becomes
v + x dv/dx = v + tan v x dv/dx = tan v
Separate and integrate
cot v dv = dx/x
ln|sin v| = ln|x| + ln c
sin v = cx
sin(y/x) = cx, where c is an arbitrary constant.
(Check: y = x sin⁻¹(cx) gives dy/dx = y/x + cx/√(1 − c²x²) = y/x + tan(y/x) ✓.)
✓ Answer checked by Claude Opus 5.5 (high effort)
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