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**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 a

_{k}x

^{k}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 a

_{k}x

^{k}

y' = [tex]\sum[/tex] from 1 to infinity of ka

_{k}x

^{k-1}

y'' = [tex]\sum[/tex] from 0 to infinity of (k+2)(k+1)a

_{k+2}x

^{k}

Substituting into the ODE I obtained,

4[tex]\sum[/tex] from 0 to infinity of (k+2)(k+1)a

_{k+2}x

^{k+1}+ 6[tex]\sum[/tex] from 1 to infinity of ka

_{k}x

^{k-1}+ [tex]\sum[/tex] from 0 to infinity of a

_{k}x

^{k}

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?