Homogenous Differential Equation

lindah · 2011-09-22 09:27:47

Hi guys,

I am not sure how to approach this question, so would like to ask for hints on approaching this rather than the answer at this point.

In terms of classes, we have not worked with Chebyshev or Legendre polynomials.
It looks like it could be classified as a Euler equation with a solution

for which I then could use reduction of order to find

May I ask if this would be a correct approach?

bobbym · 2011-09-22 09:49:56

Hi;

Is this the DE for the Legendre Polynomials?

lindah · 2011-09-22 10:07:27

Hi bobbym;

To be honest I am not sure It wasn't explicitly stated by the lecturer
I came across the Chebyshev and Legendre versions after searching the net for similar questions

bobbym · 2011-09-22 10:08:50

Hi lindah;

I am playing with it now and am unable to find any values that yield a polynomial.

Have you tried looking at the characteristic polynomial?

lindah · 2011-09-22 10:50:00

Hi bobbym;

Yes, I had a look, it would work well if the coefficients were constants, but since it has

makes it unusual to deal with the roots

I was thinking about using power series to attack this...?

Last edited by lindah (2011-09-22 10:52:35)

bobbym · 2011-09-22 10:55:58

Are you familiar with reduction of order? Also that is not an Euler-Cauchy equation.

lindah · 2011-09-22 11:02:25

Hi bobbym;

Yes, I am familiar with reduction of order. That would require I know at least one of the solutions?

bobbym · 2011-09-22 11:06:41

Usually they try y = x or y = x^2 something simple like that.

lindah · 2011-09-22 11:17:12

Hi bobbym;

I outlined that approach in my initial post, but I may think you are on the mark about Legendre polynomials.
I tried k and lambda = 0 and got

Polynomial!!!

bobbym · 2011-09-22 11:22:44

Hi lindah;

I am sorry that is the first thing I tried but I made a slight mistake and got complex infinity as the answer. Sorry, we could have been done a long time ago.

Hold on, there are other solutions

lambda = 0 and k = -2
lambda = 0 and k = -4
lambda = 0 and k = -6

It is possible k = { 0,-2,-4,-6,-8,...}

I believe that the above is true.

Solution to this DE is:

Which is a polynomial.

lindah · 2011-09-22 12:05:02

Hi bobbym;

Thank you for that!!!

I will have a try at laying out the generic solution and then your solution.
I'm reading up further on Legendre's solutions

bobbym · 2011-09-22 12:06:38

Hi Linda;

Let me know what you get.

Math Is Fun Forum

#1 2011-09-22 09:27:47

Homogenous Differential Equation

#2 2011-09-22 09:49:56

Re: Homogenous Differential Equation

#3 2011-09-22 10:07:27

Re: Homogenous Differential Equation

#4 2011-09-22 10:08:50

Re: Homogenous Differential Equation

#5 2011-09-22 10:50:00

Re: Homogenous Differential Equation

#6 2011-09-22 10:55:58

Re: Homogenous Differential Equation

#7 2011-09-22 11:02:25

Re: Homogenous Differential Equation

#8 2011-09-22 11:06:41

Re: Homogenous Differential Equation

#9 2011-09-22 11:17:12

Re: Homogenous Differential Equation

#10 2011-09-22 11:22:44

Re: Homogenous Differential Equation

#11 2011-09-22 12:05:02

Re: Homogenous Differential Equation

#12 2011-09-22 12:06:38

Re: Homogenous Differential Equation

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