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Series solution to Second-order ODE

1. Homework Statement

Find the first four non-vanishing terms in a series solution of the form [tex]\sum[/tex] from 0 to infinity of akxk for the initial value problem,

4xy''(x) + 6y'(x) + y(x) = 0, y(0) = 1 and y'(0) = -1/6

2. Homework Equations

3. The Attempt at a Solution

Taking the second derivative of the series solution form I obtained,

y = [tex]\sum[/tex] from 0 to infinity of akxk
y' = [tex]\sum[/tex] from 1 to infinity of kakxk-1
y'' = [tex]\sum[/tex] from 0 to infinity of (k+2)(k+1)ak+2xk

Substituting into the ODE I obtained,

4[tex]\sum[/tex] from 0 to infinity of (k+2)(k+1)ak+2xk+1 + 6[tex]\sum[/tex] from 1 to infinity of kakxk-1 + [tex]\sum[/tex] from 0 to infinity of akxk

Now, I am unsure of where to go from here. Does this become two separate series for [tex]\sum[/tex] from 0 to infinity and [tex]\sum[/tex] from 1 to infinity?
 

Avodyne

Science Advisor
1,396
85
No, it should be written as one series. Note that your series from 1 to infinity can just as well be written as from 0 to infinity, because kak is zero when k=0.
 

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