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! :)