Differential Equation - Series - Recurrence Relation

So the recurrence relation for even terms would be: (an-32an-ann(n-1))/(16(2n+2)(2n+1))=a2n+2 and for odd terms would be: (an-32an-ann(n-1))/(16(2n+3)(2n+2))=a2n+3. In summary, the question asks for the recurrence relation for even and odd terms in the form of a2k+2 and a2k+3. This can be obtained by substituting 2n+1 and 2n into all n's respectively.
  • #1
mirrorx
3
0
1. (16+x2)-xy'+32y=0

Seek a power series solution for the given differential equation about the given point x0 find the recurrence relation.
So I used y=∑Anxn , found y' and y''
then I substituted it into the original equation, distributed, made all x to the n power equal to xn, made the indexes 0, and added them all up.

Then I solved for an+2 and got:

(an-32an-ann(n-1))/(16(n+2)(n+1))=an+2


The question asks for for the recurrence relation in the form of a2k+2 and a2k+3
which are supposed to be the recurrence relation for even and odd terms.

How do I put it into that format? I'm just not sure where to go from this point. Also can someone even verify if I did the first part of obtaining an+2 correctly?
 
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  • #2
g9JGvhB.png


Picture of attempted solution
 
  • #3
Never mind, I figured out the question. You can just substitute 2n+1 and 2n into all n's to get odd and even terms respectively.
 

1. What is a differential equation?

A differential equation is a mathematical equation that relates an unknown function to its derivatives (or rates of change). It is commonly used to model dynamic systems in various fields such as physics, engineering, and economics.

2. What is a series in mathematics?

A series is a sequence of numbers that are added together in a specific order. It can be infinite or finite and can either converge (approach a finite value) or diverge (approach infinity).

3. What is a recurrence relation?

A recurrence relation is a mathematical relationship that defines a sequence of numbers based on the previous terms in the sequence. It is commonly used to model and analyze recursive algorithms and dynamic systems.

4. How are differential equations, series, and recurrence relations related?

Differential equations can be solved using series, which are used to represent the solutions as a sum of infinite terms. Recurrence relations can also be used to solve differential equations by expressing the derivatives in terms of the previous terms in the sequence.

5. What are some real-world applications of differential equations, series, and recurrence relations?

Differential equations, series, and recurrence relations have various applications in fields such as physics, engineering, economics, and biology. They can be used to model and predict the behavior of physical systems, analyze financial markets, and understand population dynamics, among many other applications.

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