Series solution to Second-order ODE

[tracedinair]

Homework Statement

Find the first four non-vanishing terms in a series solution of the form $$\sum$$ from 0 to infinity of a_kx^k for the initial value problem,

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

Homework Equations

The Attempt at a Solution

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

y = $$\sum$$ from 0 to infinity of a_kx^k
y' = $$\sum$$ from 1 to infinity of ka_kx^k-1
y'' = $$\sum$$ from 0 to infinity of (k+2)(k+1)a_k+2x^k

Substituting into the ODE I obtained,

4$$\sum$$ from 0 to infinity of (k+2)(k+1)a_k+2x^k+1 + 6$$\sum$$ from 1 to infinity of ka_kx^k-1 + $$\sum$$ from 0 to infinity of a_kx^k

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

 

[Avodyne]

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 ka_k is zero when k=0.