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

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Homework Help Overview

The discussion revolves around determining the existence of an analytic solution for a differential equation, specifically one that is separable. The original poster presents a differential equation involving variables x and y, expressing difficulty in isolating y after attempting to separate the variables.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants explore the method of separating variables and express concerns about isolating y. Some question the implications of solutions that involve complex forms, such as ye^y, and how these relate to real applications.

Discussion Status

The discussion is ongoing, with participants sharing their attempts and questioning the nature of solutions to differential equations. There is an acknowledgment that not all differential equations yield analytic solutions, and some guidance has been provided regarding the interpretation of solutions in practical contexts.

Contextual Notes

Participants note the challenge of obtaining y as a function of x and the implications of encountering complex forms in solutions. The original poster's context includes a focus on separable differential equations as part of their coursework.

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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

 
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So why not separate the variable like normal?
[tex](4-y^3)dy = (4x-4x^3)dx[/tex]
 
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
 
cantdotthis said:
...

I can't get the y by itself
That's pretty common with solutions to differential equations.
 
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?
 
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.
 

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