Finding Series Solutions Using Method of Frobenius

[brasidas]

Homework Statement

Using method of Frobenius, find a series solution to the following differential equation:

[tex]
x^2\frac{d^2y(x)}{dx^2} + 4x\frac{dy(x)}{dx} + xy(x) = 0
[/tex]

Homework Equations

[tex] y(x) = \sum_{n = 0}^\infty C_{n} x^{n + s}[/tex]

The Attempt at a Solution

[tex]
y(x) = \sum_{n = 0}^\infty C_{n} x^{n + s}
[/tex]
[tex]
\frac{dy(x)}{dx} = \sum_{n = 0}^\infty C_{n} (n + s) x^{n + s - 1}
[/tex]
[tex]
\frac{d^2 y(x)}{dx^2} = \sum_{n = 0}^\infty C_{n} (n + s) (n + s - 1) x^{n + s - 2}
[/tex]

Therefore, by substituting, I get:

[tex]
x^2\frac{d^2y(x)}{dx^2} = \sum_{n = 0}^\infty C_{n} (n + s) (n + s - 1) x^{n + s}
[/tex]
[tex]
4x\frac{dy(x)}{dx} = \sum_{n = 0}^\infty 4C_{n} (n + s) x^{n + s}
[/tex]
[tex]
xy(x) = \sum_{n = 0}^\infty C_{n} x^{n + s + 1} = \sum_{n = 1}^\infty C_{n - 1} x^{n + s} \rightarrow n + 1 = m \leftrightarrow n = m - 1, n \geq 0, m \geq 1
[/tex]

Combining all terms, I get:

[tex]
C_{0}((s + 0) (s + 0 - 1) + 4(s + 0))x^s + \sum_{n = 1}^\infty [C_{n} (n + s) (n + s + 3) + C_{n - 1}] x^{n + s}
[/tex]

Assuming [itex] C_{0} [/itex] is not 0, I get:

[tex]
C_{0}(s(s + 3)) = 0
[/tex]

and...

[tex]
C_{n} (n + s) (n + s + 3) + C_{n - 1} = 0
[/tex]

Now, with the assumption is that [itex] C_{0} [/itex] is not 0, I conclude that:

[tex]

s(s + 3) = 0, s = 0 , -3

[/tex]

Now... So far, so good. The problem is within the generating terms.

[tex]
C_{n} (n + s) (n + s + 3) + C_{n - 1} = 0
[/tex]

This has to be zero at all times, meaning:

[tex]
C_{n} (n + s) (n + s + 3) = - C_{n - 1}
[/tex]

Therefore:

[tex]
C_{n} = - \frac{C_{n - 1}}{(n + s) (n + s + 3)}
[/tex]

So what's the problem? You see, if we assume s = -3, and [itex] C_{0} [/itex] is not 0, then we got a problem at [itex] n = 3, s = -3[/itex] as that will mean the whole equation will explode. This means [itex] C_{0} [/itex], [itex] C_{1} [/itex], [itex] C_{2} [/itex] are all zero, with no information about [itex] C_{3} [/itex]

Am I doing it right? I am having my doubts.

 

[Mark44]

Shouldn't your DE be
[tex]x^2\frac{d^2 y(x)}{dx^2} + 4x\frac{d y(x)}{d x} + xy(x) = 0 [/tex]
?

IOW, the 2nd derivative in the first term, and derivatives intead of partial derivatives?

Take a look at this wikipedia article - http://en.wikipedia.org/wiki/Frobenius_method

 

[brasidas]

Shouldn't your DE be
[tex]x^2\frac{d^2 y(x)}{dx^2} + 4x\frac{d y(x)}{d x} + xy(x) = 0 [/tex]
?

IOW, the 2nd derivative in the first term, and derivatives intead of partial derivatives?

Take a look at this wikipedia article - http://en.wikipedia.org/wiki/Frobenius_method

I wasn't done typing the problem, and my attempt at it.

Trivial errors are all fixed by now. That aside...

I still don't see how much sense I can get out of the situation above.

 

[brasidas]

I guess this problem doesn't need any more attention.

My understanding is that Method of Frobenius may be of help to find a solution to the DEQ, but it may not be able to provide all the solutions.

In this case, s = -3 doesn't provide anything useful, for instance. s = 0 is the only sensible choice, in other words.