# Problem with an ODE

1. Feb 18, 2005

### 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...

2. Feb 18, 2005

### 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.

3. Feb 18, 2005

### 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!

4. Mar 2, 2005

### 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.