Question

solve x dy/dx + y = xy^3​

Answer 1

Making the change of variable y=1z we have the new version

dydx+y−xy3=0→x−z2+zz'z3=0 or

x−z2+zz'=0

Now calling ξ=z2 we have

x−ξ+12ξ'=0

Solving for ξ we obtain easily

ξ=12+x+Ce2x=z2 then

z=±√12+x+Ce2x=1y then finally

y=±1√12+x+Ce2x