# Thread: Power Series Solutions of ODE's

1. ## Power Series Solutions of ODE's

consider the differential equation (1+x^2)y''-6y=0.
show that x=0 is an ordinary point of this equation - which i have done.

determine the interval in x for which the equation has a convergent solution of the form y=sigma(akx^k).

Show that the recurrence relation is
ak+2=-[(k-3)/(k+1)]*ak

and hence, determine the first six non-zero terms of the series solution.

2. Originally Posted by nahal

consider the differential equation (1+x^2)y''-6y=0.
show that x=0 is an ordinary point of this equation - which i have done.

determine the interval in x for which the equation has a convergent solution of the form y=sigma(akx^k).

Show that the recurrence relation is
ak+2=-[(k-3)/(k+1)]*ak

and hence, determine the first six non-zero terms of the series solution.

these problems are a pain to type out with LaTex, so i will give you some references. tell me if they help

see here

and here

and also here

just to start you off. we are expanding around $x_0 = 0$, so we assume a solution of $y = \sum_{k = 0}^{\infty}a_kx^k$

so, $y' = \sum_{k = 1}^{\infty}ka_kx^{k - 1}$

and, $y'' = \sum_{k = 2}^{\infty}k(k - 1)a_k x^{k - 2}$

now your problem is: $(1 + x^2)y'' - 6y = 0$

$\Rightarrow y'' + x^2y'' - 6y = 0$

now plug in the series forms for y and y'' and continue as you see in the links i gave you

3. Thank you for all your help, it was well appreciated.

There werew some very useful links that were provided.

4. Originally Posted by nahal
Thank you for all your help, it was well appreciated.

There werew some very useful links that were provided.
you're welcome.

i suppose you solved the question ok then?