5.38, §2 Solution

xy' + y = x y³

It is a Bernoulli equation since it has terms linear in y and a power of y.

x y^-3 y' + y^-2 = x

Put u=y^-2:

Solution of the homogeneous equation:

Solution of the inhomogeneous equation:

u = C(x) x²

into the inhomogeneous equation:

Solution:

For C₀=0 :

For x=-2/C₀, y is infinite.

For C₀<0:

For C₀>0:

Total: