Can you prove a differential equation has no analytical solution?

[chener]

Can you prove a differential equation has no analytical solution?

Teach me ,please!

Thank you a lot!

 

[Erland]

Is the problem only to find one differential equation which has no analytical solution? If so, it is easy:

Take for example ##y'=2|x|##. This has the general solution ##y=sgn(x) x^2 + C##. None of these solutions is analytic, since they are not twice differentiable at 0.

 

[HallsofIvy]

But any such differential equation will have a solution that is analytic on some set. Chener, please give us more information on exactly what you mean.

 

[Erland]

But any such differential equation will have a solution that is analytic on some set. Chener, please give us more information on exactly what you mean.

Not if we take ##y'=f(x)##, where ##f(x)## is a function which is continuous everywhere but differentiable nowhere. The general solution of this equation is ##y=F(x)+C##, where ##F## is an antiderivative of ##f##. All these solutions are analytic nowhere, since they are twice differentiable nowhere.

But I agree that Chener must tell us exactly what he means.

 

[chener]

Sorry my English is poor !
Let me think about how to express

 

[chener]

Given a Differential Equation.

Is there Analytic Solutions or not?

And how do we know that?example: Given the equation y''=-y-y'+(1+|y|)*sin(t)

,how to prove a it has no analytical solution?Teach me ,please!

Thank you a lot!
Analytic Solution is resovle from Analytic Method.

 

[bigfooted]

Maybe you want to ask if there are methods to prove that an equation has a closed form expression as the general solution?
Let us know if this is what you mean:
http://en.wikipedia.org/wiki/Closed-form_expression

 

[chener]

Yes ,this is my mean! thanks!

 

[Stephen Tashi]

analytical solution!

You must explain what you mean by "analytical solution".

The term "analytic function" has a technical meaning from the theory of complex variables. It is a function that can be expanded in a power series.

You might be using the phrase "analytical solution" to mean a function that can be written down as a finite string of symbols - such as a finite sum of products of polynomials, trig functions etc. The phrase "closed form expression" is used to describe such a function.

Whether the solution to a problem in calculus has a solution that is a "closed form expression" is a question about how strings of symbols can be manipulated.

Whether the solution to a problem in calculus is an "analytic function" is a problem of [itex] \epsilon [/itex] and [itex] \delta [/itex] reasoning.

 

[slider142]

Your question, now that it has been clarified to mean "Is there a method or theorem that can definitively tell whether a given differential equation has a closed-form expression in terms of some collection of elementary functions?" is the subject of Differential Galois Theory. This theory started out with Liouville's ideas on being able to tell when an integral had a closed form expression and has since expanded from there. It is still an active area of research, as far as I know. Here is a taste of how the theory proceeds, starting with the simplest task of a separable differential equation, which is just an integral: http://www2.maths.ox.ac.uk/cmi/library/academy/LectureNotes05/Conrad.pdf .