The solution of differential equationyy′=x(y2x2+f(y2/x2)f′(y2/x2)) is

[a] The given equation can be written as$yxdydx=\left\{y2x2+f\left(y\frac{2}{x}2\right)f\prime \left(y\frac{2}{x}2\right)\right\}$The above equation is a homogeneous equation.$Pu\in gy=vx,we\ge t$v[v+xdvdx]=v2+f(v2)f′(v2)$\text{or}vxdvdx=f\left(v2\right)f\prime \left(v2\right)\left(variab\le separab\le \right)$or 2vf′(v2)f(v2)dv=2dxx$Now,\int egrat\in g\perp hsides,we\ge t$logf(v2)=logx2+logc[logc=constant]$\text{or}\log f\left(v2\right)=\log cx2\text{or}f\left(v2\right)=cx2\text{or}$f(y2/x2)=cx2