Differential equation

[Sriman Dutta]

Hi everyone,

 

Just like the Abel-Ruffini theorem limits the solution to polynomial equations upto fourth degree, is there any such check on the solutions of a differential equation? Can a complete general solution be framed for any n-th order differential equation ?

 

Thanks in advance

[studiot]

Well yes and no.

 

The Abel-Ruffini theorem refers to polynomials; these are a type of linear equations in linear algebra.

 

In the same way a fourth or nth order linear differential equation can be written in a (linear) polynomial in the D operator

 

eg (aD⁴ + bD³ + cD² + dD + e)(x) = 0 : a, b c, d, e are constants; D is the D operator.

 

Has a general solution related to the solution of the fourth order polynomial.

 

Here are lots of pdfs and youtubes on solving linear differential equations by the D operator.

 

https://www.google.co.uk/?gws_rd=ssl#q=operator+method+for+linear+differential+equations

 

 

But general fourth order differential equations may contain terms that preclude this or may be nonlinear

 

in which case the answer is no.

Edited May 11, 2017 by studiot

[Sriman Dutta]

Thanks for the reply.

[Country Boy]

Hi everyone,

 

Just like the Abel-Ruffini theorem limits the solution to polynomial equations upto fourth degree, is there any such check on the solutions of a differential equation? Can a complete general solution be framed for any n-th order differential equation ?

 

Thanks in advance

 

 

 

This is not true. The Abel-Ruffini theorem simply asserts that a polynomial equation, of degree greater than 4, may have solutions that cannot be written in terms of radicals.

Every polynomial equation, of degree n, has exactly n solutions in the complex numbers, counting multiple roots.

 

Every such differential equation has such a "complete general solution". It may, of course, be very difficult to write down.

Edited May 20, 2017 by Country Boy

[Mark O'Connor]

Yeah, I totally agree my affiliate.  You know, it's one of those things were you just can't tell, so keep at it and get that degree! Don't let anyone put you down my man! What a guy. 

Edited March 20, 2018 by Mark O'Connor
Error while typing.

[Conor Cowley]

Have you tried turning it on and off again? lol jk bro, Hope all goes well in your degree pal.

 

[swansont]

6 hours ago, Mark O'Connor said:

Yeah, I totally agree my affiliate.  You know, it's one of those things were you just can't tell, so keep at it and get that degree! Don't let anyone put you down my man! What a guy. 

 

6 hours ago, Conor Cowley said:

Have you tried turning it on and off again? lol jk bro, Hope all goes well in your degree pal.

 

!

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