Frobenius method indical roots differing by a postive integer

[tphysicsb]

Homework Statement

For what values of the constant K does the differential equation
y"-(1/4 +K/x)y=0 (0<X< infinity)

have a nontrival solution vanishing at x=0 and x= infinity ?

Homework Equations

Hints that were suggested for my prof were:
For large x assume K/x is small and drop this term to obtain and approximate solution valid for large X. One solution found will be singular at infinity (cast out the solution with the singularity at infinity. Obtain a 2nd solution using reduction of order and then solve the resulting equation using a power series. One series for the larger root will yield a series that can be truncated for certain values of K

The Attempt at a Solution

I followed his hints and assumed X was larger and solved the resulting 2nd order ode with constant coffeicents. I obtained

y= C1e^-x/2 and C2 e^x/2


I then cast out the solution e^x/2 and used the reduction or order method
y=v(x)P(x) where P(x) = e^-x/2
after finding y' y'' I plugged them into the original 2nd order ode and after performing all the algebraic manipulation I ended up with a new 2nd order ode

v"-v'-(K/X)v=0

I then used the frobenius method to solve this diff. eq.

I ended up with indical roots of s=0; s=1

And the following recursion relationships

S=0: Cn+1=(n+k)/(n+1)(n) Cn for n=0,1,2...

S=1: Cn+1= (n+1+k)/(n+2)(n+1) Cn for n=0,1,2...

for the S=0 relationship, when I plug in n=0 I am dividing by zero.

I have read in other resources that their is a problem at occurs when your indical roots differ by ... I want to say postive Interger and I realize that the frobenious method only guaranties one solution, but I am not what the next step is in order to find the 2nd solution.



*(This is my first time posting so on a side note if anyone that is familiar could provide me with a reference on using the latex equation editor it would be greatly appreciated)

 

[mighty2000]

Hello,

Thank you for your detailed post. It seems like you have made good progress in solving the differential equation. However, as you mentioned, there is an issue with the recursion relationships when plugging in n=0. This is because the series solution with s=0 is a singular solution and cannot be used for all values of K. This is where the second solution with s=1 comes in.

To find the second solution, you can use the method of variation of parameters. This method involves finding a particular solution for the equation and then using it to construct the general solution. In this case, you can use the series solution for s=0 as the particular solution. This will give you a general solution of the form:

y=C1e^-x/2 + C2e^x/2 + C3e^-x/2lnx

Where C3 is a constant to be determined. You can then use the initial conditions at x=0 and x=infinity to solve for C1 and C2. For example, if the solution needs to vanish at x=0, then C1=0.

I hope this helps. Good luck solving the differential equation!

As for using the LaTeX equation editor, there are many online resources available such as tutorials and forums where you can ask for help. You can also refer to the LaTeX documentation for more information.