# Thread: Find two power series solutions.

1. ## Find two power series solutions.

ok i have a problem, that i cant seem to get started on...

so here is the problem..

after substituting $y=\sum_{n=0}^{\infty}c_{n}x^{n}$ into a differential equation, the resulting equation is given below. find two power series solutions. list at least 5 non-zero terms in each sequence before using the....

$2c_{0}+2c_{2}+\sum_{k=1}^{\infty}[(3k+2)c_{k}+k(k+1)c_{k+1}-(k+1)(k+2)c_{k+2}]x^{k}=0$

im thinking that i just substitute different values for k to find the first five terms? is this right? or am I off in my approach?

any thoughts or assistance would be greatly appreciated...

2. Each coefficient of x^k is equal to 0

k=0 => 2c_0+2c_2=0
k=1 => 5c_1+2c_2-6c_3 = 0
k=2 => 8c_2+6c_3-12c_4 = 0
and so on

You can see that choosing the first 2 coefficients gives all the coefficients

3. Originally Posted by running-gag
Each coefficient of x^k is equal to 0

k=0 => 2c_0+2c_2=0
k=1 => 5c_1+2c_2-6c_3 = 0
k=2 => 8c_2+6c_3-12c_4 = 0
and so on

You can see that choosing the first 2 coefficients gives all the coefficients
ok i dont see how the x^k on the end will cancel out for the different values of k, maybe at k=0, but as k values increase im not sure they will cancel out...

can you please explain further?