(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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]

2. Relevant equations

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

3. 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.

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# Homework Help: Method of Frobenius (I)

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