求解微分方程dy/dx=-(y/x)(2x^3-y^3)/(2y^3-x^3)

Question

求解微分方程
dy/dx = -(y/x) (2x^3 - y^3)/(2y^3 - x^3)

Answer 1

接：设y=xt,则dy/dx=t+xdt/dx
代入原方程,得t+xdt/dx=-t(2-t³)/(2t³-1)
==>xdt/dx=t(t³+1)/(1-2t³)
==>(1-2t³)dt/[t(t³+1)]=dx/x
==>[1/t-1/(t+1)-(2t-1)/(t²-t+1)]dt=dx/x
==>∫dt/t-∫d(t+1)/(t+1)-∫d(t²-t+1)/(t²-t+1)=∫dx/x
==>ln│t│-ln│t+1│-ln│t²-t+1│=ln│x│+ln│C│ (C是积分常数)
==>ln│t/[(t+1)(t²-t+1)]│=ln│Cx│
==>t/[(t+1)(t²-t+1)=Cx
==>(y/x)/[(y/x+1)(y²/x²-y/x+1)]=Cx
==>x²y/[(y+x)(y²-xy+x²)]=Cx
==>xy/(x³+y³)=C
==>xy=C(x³+y³)
故原方程的通解是xy=C(x³+y³) (C是积分常数).