# Bernoulli equation

1. Oct 8, 2005

### asdf1

can someone explain how to solve the bernoulli equation? i'm having a hard time understanding...

2. Oct 8, 2005

### mezarashi

The Bernoulli equation is an energy conservation equation for fluid kinetics. In what way are you having difficulty solving it?

3. Oct 8, 2005

### saltydog

You mean:

$$y^{'}+P(x)y=Q(x)y^n$$

The key to solving this is to recognize the differential form:

$$y^{-n}dy$$

and what, when differentiated, gives this. Well that would be:

$$\frac{1}{1-n}y^{1-n}$$

Hey, I know it's not easy. They catch me in here all the time with differential forms.

Ok then so we'll divide by $y^n$ up there in the first equation and take the differential form:

$$y^{-n}dy+Py^{1-n}dx=Qdx$$

Alright then,so that's what we have right, the differential $y^{-n}dy$.

So, let:

$$z=y^{1-n}$$

and then substitute the differential form of this into the original equation. Here's the first part:

We got:

$$y^{-n}dy+Py^{1-n}dx=Qdx$$

So the $y^{-n}dy$ part would just be:

$$\frac{1}{1-n}dz$$

Do the rest and then get a first-order ODE in terms of z and x.

Last edited: Oct 8, 2005
4. Oct 8, 2005

### asdf1

hmm... so the key is to try to get the non-linear equation into a linear equation...
saltydog, thank you very much for explaining it to me!!! :)