Find the power series?

Question

Find the power series in x for the general solution of (1+x^2)y"+6xy'+6y=0.

 

Here's my work:

 

Ly=(1+x^2)y"+6xy'+6y

y=Σa_nxⁿ from n=0 to infinity

y'=Σna_nx^n-1 from n=1 to infinity

y"=Σn(n-1)a_nx^n-2 from n=2 to infinity

Now what? I'm stucked in here. Please show all your work step by step.

Answer 1

Hi Yang,

 

I looked at this question and it's pretty straightforward (as far as problems like this go, which means it is messy).

 

What you started to do is fine. The next step is to plug in the three power series into the differential equation, collect like terms (which requires some index shifting), set each coefficient equal to zero, then start solving for the original power series coefficients.

 

Rather than supply all of the details (which would wear out my fingers and patience), I worked this out on paper and will show you the key results. You can then reply back to me if anything needs to be clarified.

 

When I substituted the series into the DE and simplified, I got the following terms:

 

(a₂ + 6a₀) + (6a₃ + 12a₁)x + ∑_n=2^∞ [(n+2)(n+1)a_n+2 + (n+3)(n+2)a_n]xⁿ

 

Since this must equal zero, all coefficients for this series must equal zero over the interval of convergence.

 

When you do this, you get the following equations:

 

a₂ = -6a₀,

a₃ = -2a₁,

 

and the recurrence relation

 

a_n+2 = -(n+3)/(n+1)a_n, for n≥2.

 

What you must do then is to start plugging in numbers from 2 onward into the recurrence relation, and write each coefficient in terms of a₀ or a₁.

 

Here is what I got (you should check this because I do not do this every day):

 

a₄ = 10a₀

a₅ = 3a₁

a₆ = -14a₀

a₇ = -4a₁

a₈ = 18a₀

a₉ = 5a₁

 

etc.

 

Put these in the original series and collect the terms with a₀ together and separate the terns with a₁ from them.

 

What you will get is a linear combination of two power series:

 

y₁(x) = 1 - 6x² + 10x⁴ - 14x⁶ + 18x⁸ - ...

 

and 

 

y₂(x) = x - 2x³ + 3x⁵ - 4x⁷ + 5x⁹ - ...

 

The solutions then have the form y(x) = a₀y₁(x) + a₁y₂(x). This will be the general solution. I did not bother to try writing the nth term of each series using a closed formula, which you may be expected to do by your instructor. This is not always easy to do and I have said enough already, so I will leave it like this. I also did not find the radius or interval of convergence as you did not ask for these, but they are important aspects of the solution.

 

If you need more help, you know how to reach me.

 

Have fun!