求微分方程y'+y/x=x^2y^6的通解

Question

求微分方程y'+y/x=x^2y^6的通解

Answer 1

令z=1/y^5,则y'=(-y^6/5)z'
代入原方程,并整理得z'/5-z/x=-x².(1)
∵方程(1)的齐次方程是z'/5-z/x=0
==>dz/z=5dx/x
==>ln│z│=5ln│x│+ln│C│ (C是积分常数)
==>z=Cx^5
∴此齐次方程的通解是z=Cx^5
于是,设方程(1)的解为z=C(x)x^5 (C(x)是关于x的函数)
∵代入方程(1)得(C'(x)x^5+5C(x)x^4)/5-C(x)x^4=-x²
==>C'(x)x^5=-5x²
==>C'(x)=-5/x³
==>C(x)=5/(2x²)+C (C是积分常数)
==>z=5x³/2+Cx^5
∴方程(1)的通解是z=5x³/2+Cx^5
故原方程的通解是1/y^5=5x³/2+Cx^5 (C是积分常数).