設(shè)y'=p,則y''=pdp/dy
代入原方程得pdp/dy-p³-p=0 ==>p(dp/dy-p²-1)=0
∴p=0,dp/dy-p²-1=0
∵當(dāng)p=0時(shí),有dy/dx=0 ==>y=C (C是積分常數(shù))
∴經(jīng)檢驗(yàn)y=C是原方程的解
∵當(dāng)dp/dy-p²-1=0時(shí),有dp/dy=p²+1
==>dp/(p²+1)=dy
==>arctanp=y+C1 (C1是積分常數(shù))
==>p=tan(y+C1)
==>dy=tan(y+C1)dx
==>cos(y+C1)dy/sin(y+C1)=dx
==>d(sin(y+C1))/sin(y+C1)=dx
==>ln│sin(y+C1)│=x+ln│C2│ (C2是積分常數(shù))
==>sin(y+C1)=C2e^x
∴sin(y+C1)=C2e^x也是原方程的解
故原微分方程的通解是y=C,或sin(y+C1)=C2e^x （C,C1,C2是積分常數(shù)）.