Solve Power Series: (x^2)(y")+y=0

[momen salah]

hi guys it's me you helped me last time in my bonus problem(thank's for that) i need help again naw please it's a hard problem for me :

solve using power series:
(x^2)(y")+y=0

after solving it i stopped at :

an[n^2-n+1]=0

 

[exk]

If you are trying to solve this equation around the point $x_0=0$ you will run into a problem since that is a singular point of that equation.

 

[Crosson]

Are you using the method of Frobenius?

Here is the general solution in terms of elementary functions:
$$\sqrt{x} \cos \left(\frac{1}{2} \sqrt{3} \log (x)\right) C_1+\sqrt{x}<br /> \sin \left(\frac{1}{2} \sqrt{3} \log (x)\right) C_2$$

 

[momen salah]

i don't know we only learned how to solve de's when there are no singularities

 

[exk]

You will have to use Frobenius method for this particular problem.
You can find more info with an example here: http://en.wikipedia.org/wiki/Frobenius_method

 

[momen salah]

how can i use this Frobenius method

 

[uart]

how can i use this Frobenius method

It's quite an interesting problem, I just took the time to solve it now. Normally with a series solution you expect to find a recursion relation for the coefficients, but this one is quite different. Here, as I'm sure you've already discovered (and I'm guessing where you got stuck), there is no recursion relation and at first it may appear that all the coefficients are forced to zero.

If you apply the Frobenius method you should end up with something in the form of,

$$P(r+k) \, \, a_k = 0$$, for each k, where P(.) is a polynomial.

No recursion relation, and all $$a_k$$ must be zero except for at most a finite number corresponding to the zeros of P(.).

If you got this far and it all seemed wrong then don't depair, you're on the right track. Here are some hints to finish it off.

1. Let $$r + k = \omega$$ (or whatever) and solve $$P(\omega) = 0$$

2. If you get complex roots then keep going regardless. (Don't worry as later in the solution you can still restrict the remaining coefficients to force y to be a real function).

3. Don't forget that the complex exponential $$x^{i \beta}$$ can be rewritten as $$e^{i \beta \log(x)}$$.

Follows those hints and you'll get Crosson's solution with surprisingly little effort.

 

[Big-T]

Why not "guess" a solution of the form x^r, and end up with solutions $$x^{1/2 \pm i\sqrt{3}/2}$$, which are essentially the same as Crosson ended up with?

 

[uart]

There's nothing wrong with using an inspired guess (and verify) to solve a DE, but in this instance the OP did explicitly say that he was asked to solve it using a power series method.

 

[Big-T]

Alright, thanks!