Solve Bernoulli Equation to Understanding

[asdf1]

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

 

[mezarashi]

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

 

[saltydog]

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

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.

 

[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! :)