Solving ODE with Paper and Pencil: xy^2 - y = \frac {dy} {dx}

[Neoma]

\frac {dy} {dx} = x y^2 - y

I used Mathematica's DSolve function and found the correct answer:
y(x) = \frac {1} {1 + x + C e^{x}}

However, I don't have any idea what method to use to solve it with pencil and paper...

 

[dextercioby]

I cheated,i know;i probably wouldn't have seen it,if you hadn't provided the answer.

Make the substitution:

y(x)=\frac{1}{u(x)}

I believe you'll like the ODE that comes out.

Daniel.

 

[Neoma]

I wasn't familiar with the substitution method yet, so I looked it up after reading your post and it looks quite elegant :)

Thanks!

 

[Jayboy]

Just to put this problem in a general context, it's form is:

\frac {dy} {dx} = a(x) y + b(x) y^p

Which is a Bernoulli ODE (or a Ricatti with no constant term).

The substitution:

u(x) = y^{1-p}

reduces this to a first order linear ODE which can be solved in the usual way via an integrating factor.