Pointers for the solution of 2nd order DE with variable coefficient

[crawf_777]

Hi,

I am looking at the following 2nd order DE with variable coefficient:

y''(x)-(1/x+7)y(x)=0

I would be grateful for any help in regard to methods which may be applied to such an equation.

Many thanks in advance!

C.

 

[HallsofIvy]

Typically such an equation can be solved by looking for a series solution.

The first thing I would do is multiply both sides by x:
$$x\frac{d^2y}{dx^2}- (7x+ 1)y= 0$$

Because of that x in the leading term you will need to use "Frobenius' method"- we look for a solution of the form
$$y(x)= \sum_{n=0}^\infty a_nx^{n+c}$$
where c is not necessarily a positive integer.

Then
$$y'(x)= \sum_{n=0}^\infty (n+c)a_nx^{n+c- 1}$$
$$y''(x)= \sum_{n= 0}^\infty (n+c)(n+c-1)x^{n+c- 2}$$

Put that into the equation to get sums of powers of x equal to 0.

Choose c by looking at the coefficient of $x^0= 1$ and asserting that $a_x$ (n=0) is not 0.

Once you know c, combine like powers and set the coeficients equal to 0. You will get a recursive formula for the coefficients, $a_n$.

 

[crawf_777]

Thanks for your help HallsofIvy! Much appreciated.

C.

 

[JJacquelin]

Hello !

This ODE can be analytically solved. The closed form for the general solutions involves the Kummer function and the Tricomi function ( i.e. the confluent hypergeometric functions)

 

[crawf_777]

Hi JJacquelin,
Thanks for your pointer; I got a series solution together ok, though it seems to blow up strangely. Anyway, I would certainly be interested in seeing the analytical form of the solution to said de. I’ll have a look at those functions that you suggested.
Cheers,
C

 

[JJacquelin]

Hello crawf_777

The analytical solution is :

 

[crawf_777]

Hi JJacquelin,
Thanks for that! Very much appreciated!
C