# Series solution to Second-order ODE

#### tracedinair

1. Homework Statement

Find the first four non-vanishing terms in a series solution of the form $$\sum$$ 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 = $$\sum$$ from 0 to infinity of akxk
y' = $$\sum$$ from 1 to infinity of kakxk-1
y'' = $$\sum$$ from 0 to infinity of (k+2)(k+1)ak+2xk

Substituting into the ODE I obtained,

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

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?

Related Calculus and Beyond Homework Help News on Phys.org

#### 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 kak is zero when k=0.

### Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving