Question

dy/dx=y+y^3的通解

Answer 1

设z=1/y²,则y²=1/z,y'=-z'/(2yz²)

代入原方程得-z'/(2yz²)=y+y³

==>z'=-2(z+1)

==>dz/(z+1)=-2dx

==>ln│z+1│=-2x+ln│C│ (C是积分常数)

==>z+1=Ce^(-2x)

==>1/y²=Ce^(-2x)-1

==>[Ce^(-2x)-1]y²=1

故原方程的通解是[Ce^(-2x)-1]y²=1 (C是积分常数).