How to Determine if an Analytic Solution Exists for a Differential Equation?

[cantdotthis]

Homework Statement

I can't figure out how to solve this problem. When i do it straight forward, I get a crazy complex equation. I think I need to play with it a little before I ∫ it, but I am not sure. Everything is real, and it is in my seperable diffy eg section

dy/dx=(4x-4x^3)/(4-y^3)

Homework Equations

The Attempt at a Solution

 

[Simon Bridge]

So why not separate the variable like normal?
$$(4-y^3)dy = (4x-4x^3)dx$$

 

[cantdotthis]

I did that and got
4y-(4y^4)/4=2x^2-(x^4)/4+c.
But I think that I need to make it y=F(x). I can't get the y by itself

 

[SammyS]

...

I can't get the y by itself

That's pretty common with solutions to differential equations.

 

[QuarkCharmer]

I'm having a similar problem with simple DE's that are solvable via separation of vars or using an integration factor. Many problems come up with solutions containing y in the form of ye^y. My professor said that it's fine as the solution for her tests and so on, but in a real application, what would you do?

 

[Simon Bridge]

In a real application you'd use it as the basis for a numerical solution.
Note: ye^y is not the problem - it is that ye^y=f(x), and f(x) is the problem because it can be anything.

When you can get one as a function of the other you have an analytic solution.
There are a very large number of situations where you don't get one... in fact, for an arbitrary DE it is usually the case.

working out whether an analytic solution exists is a tricky part of number theory.