Find a recurrence formula for the power series solution around t = 0

In summary: Your name]In summary, the user is struggling to find a recurrence formula for the power series solution of a given differential equation. The expert suggests a possible approach by rewriting the equation in terms of the power series solution and then comparing the coefficients of t^n on both sides. The resulting recurrence formula is a_n+2 = -(n+2)(n+1) * a_n / (n+2)^2, valid for n ≥ 0.
  • #1
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Homework Statement


Find a recurrence formula for the power series solution around t = 0 for the differential equation:

d^2 y/dt^2 + (t - 1) dy/dt + (2t - 3)y = 0


Homework Equations


y = Σn=0 to inf (a_n * t^n)
and formula to differentiate polynomials.


The Attempt at a Solution


I can't find a way to bring all the sums to the same index. I've though about the fact that I can just change which index the sum is at for some of the sums because n=0 and n=1 could cause the entire term to be 0 itself but everything I think about seems to have at least one of the sums starting from another index. My attempt at a solution is attached.
 

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  • #2


Thank you for your question. I understand your struggle in finding a recurrence formula for the power series solution of the given differential equation. Here is a possible approach that may help you.

First, let's rewrite the differential equation in terms of the power series solution:

d^2 y/dt^2 + (t - 1) dy/dt + (2t - 3)y = 0

= Σn=0 to inf (a_n * n(n-1) * t^(n-2)) + (t - 1) Σn=0 to inf (a_n * n * t^(n-1)) + (2t - 3) Σn=0 to inf (a_n * t^n)

= Σn=0 to inf (a_n * (n(n-1) + n(t-1) + 2t - 3) * t^n)

= Σn=0 to inf (a_n * (n^2 - n + nt - n + 2t - 3) * t^n)

= Σn=0 to inf (a_n * (n^2 + (nt - 2n) + (2t - 3)) * t^n)

= Σn=0 to inf (a_n * (n^2 + (n - 2) * t + (2t - 3)) * t^n)

Now, we can compare the coefficients of t^n on both sides:

a_n * (n^2 + (n - 2) * t + (2t - 3)) = 0

This equation must hold for all values of n, so we can set n^2 + (n - 2) * t + (2t - 3) = 0.

Solving this quadratic equation for t, we get two roots: t = 1 and t = -2.

Therefore, the recurrence formula for the power series solution is:

a_n+2 = -(n+2)(n+1) * a_n / (n+2)^2

This formula is valid for n ≥ 0.

I hope this helps you in finding the solution. Let me know if you have any further questions.


 

What is a recurrence formula for a power series solution?

A recurrence formula for a power series solution is a mathematical expression that allows us to find the coefficients of a power series expansion. It relates the coefficients of one term to the coefficients of previous terms in the series.

How do I find a recurrence formula for a given power series?

To find a recurrence formula for a power series, you can start by expressing the series in terms of its coefficients. Then, you can manipulate the series using algebraic techniques to derive a formula that relates the coefficients of one term to the coefficients of previous terms.

Why is it important to find a recurrence formula for a power series solution?

A recurrence formula allows us to efficiently find the coefficients of a power series without having to manually calculate each term. This is especially useful when dealing with large or complex series, as it can save time and effort in finding the solution.

Can a recurrence formula be used for any power series solution?

Yes, a recurrence formula can be used for any power series solution as long as the series is convergent and the coefficients can be expressed in a pattern or sequence.

Are there any alternative methods for finding a power series solution besides using a recurrence formula?

Yes, there are other methods such as using the Cauchy product, using the ratio or root test, or using the Maclaurin series expansion. However, a recurrence formula is often the most efficient and straightforward method for finding the coefficients of a power series solution.

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