Question

求（y+x^3)dx=2xdy的通解

Answer 1

设u=y+x^3

有u'=y'+3x^2,因此y'=u'-3x^2

代入方程u-2xu'+6x^3=0

再设v=u/(x^(1/2))

有v'=u'/(x^(1/2))-u/(2x^(3/2))=(2xu'-u)/(2x^(3/2))=3x^(3/2)

解得v=6/5*x^(5/2)+C

故u=6/5*x^3+Cx^(1/2)

最后y=1/5*x^3+Cx^(1/2)