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

[Neoma]

[tex]\frac {dy} {dx} = x y^2 - y[/tex]

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

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:

[tex] y(x)=\frac{1}{u(x)} [/tex]

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:

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

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

The substitution:

[tex] u(x) = y^{1-p} [/tex]

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