dy/dx=y3/(x3+xy2)求通解

Question

dy/dx=y3/(x3+xy2)求通解

Answer 1

∵令y=xt,则dy/dx=xdt/dx+t
代入原方程,化简得 xdt/dx+t=t^3/(1+t^2)
==>(1/t+t)dt+dx/x=0
==>ln│t│+t^2/2+ln│x│=ln│C│ (C是常数)
==>xte^(t^2/2)=C
==>ye^(y^2/(2x^2))=C
∴原方程的通解是ye^(y^2/(2x^2))=C.