Determine the general solution for the non-linear homogeneous ODE y′=yx+sin(yx)y' = \frac{y}{x} + \sin\left(\frac{y}{x}\right)y′=xy+sin(xy).
ln∣tan(y2x)∣=ln∣x∣+C\ln|\tan\left(\frac{y}{2x}\right)| = \ln|x| + Cln∣tan(2xy)∣=ln∣x∣+C
ln∣csc(yx)−cot(yx)∣=ln∣x∣+C\ln|\csc\left(\frac{y}{x}\right) - \cot\left(\frac{y}{x}\right)| = \ln|x| + Cln∣csc(xy)−cot(xy)∣=ln∣x∣+C
cos(yx)=xln∣x∣+C\cos\left(\frac{y}{x}\right) = x \ln|x| + Ccos(xy)=xln∣x∣+C
sin(yx)=1x+C\sin\left(\frac{y}{x}\right) = \frac{1}{x} + Csin(xy)=x1+C