Question

xdy/dx-y+xsin(y/x)=0

Answer 1

x \tan(\frac{y}{2x}) = A

Answer 2

The given differential equation is homogeneous. We use the substitution $y=vx$, which transforms the equation into a separable one. After separating variables $v$ and $x$, we integrate both sides. The integration of $\frac{1}{\sin v}$ yields $\ln|\tan(\frac{v}{2})|$, and integration of $\frac{1}{x}$ yields $\ln|x|$. Combining these terms and substituting back $v = \frac{y}{x}$ gives the general solution.