Differential Equations For Solving A Recursive equation

In summary, the conversation discusses a proof of a recursive equation for the function c_n(a). The recursive equation is derived from an infinite-order linear differential equation, and the solution is a function of a, expressed as c_n(a) = Ce^(sn), where s is a function of a. This method of solving recursive equations is useful in various applications.
  • #1
MAGNIBORO
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Hi, i have a question about a proof of some recursive equation,
the function is
$$c_{n}(a)=\int_{0}^{\pi } \frac{cos(nx)-cos(na)}{cos(x)-cos(a)}$$
whit ##n\in \mathbb{N}## and ##a\in \mathbb{R}## .

whit some algebra is easy to see ##c_{0}(a)=0## and ##c_{1}(a)=\pi##

and the recursive equation

$$c_{n+1}(a)+c_{n-1}(a) -2 \, cos(a) \, c_{n}(a) =0$$

Then the author says that this is a differential equation and solves it (making ##c_{k}(a)=Ce^{sk}##)

I don't understand , i mean this is a recursive relation not a differencial equation.
so every recursive relation can be solve with a differencial equation?
thanks
 
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  • #2
MAGNIBORO said:
$$c_{n+1}(a)+c_{n-1}(a) -2 \, cos(a) \, c_{n}(a) =0$$

I don't understand , i mean this is a recursive relation not a differencial equation.

For a fixed value of ##a## you have one recursive equation. Each such recursive equation has a solution that is a function of "##a##".

Then the author says that this is a differential equation and solves it (making ##c_{k}(a)=Ce^{sk}##)
Does that notation indicate that the right hand side of the equation is independent of ##a##?

Whether it does or not, I can see that the author would use words to indicate that he is doing more than solving one recursive equation. We have one recursive equation for each different value of ##a##. I agree that it's confusing to call this infinite family of recursive equations a "differential equation".
 
  • #3
Stephen Tashi said:
For a fixed value of ##a## you have one recursive equation. Each such recursive equation has a solution that is a function of "##a##".Does that notation indicate that the right hand side of the equation is independent of ##a##?

Whether it does or not, I can see that the author would use words to indicate that he is doing more than solving one recursive equation. We have one recursive equation for each different value of ##a##. I agree that it's confusing to call this infinite family of recursive equations a "differential equation".
hi, i was searching and i see that the recursive equation can solve by the formula ##c_{1} \, \alpha ^{n} + c_{2}\, \beta ^{n}## where alpha and beta are roots of a polynomial. the autor did something similar but instead of making ##c_{n} = h^{n}## , he makes ##c_{n} = C e^{sn}##.
I leave you the part of the paper
https://gyazo.com/472073d82147bdebf094c2c66cb12f16 [Broken]
thanks
 
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  • #4
MAGNIBORO said:
I leave you the part of the paper
https://gyazo.com/472073d82147bdebf094c2c66cb12f16 [Broken]

He make ##s## a function of ##a##.

He does not call the equation in question a "differential equation". He calls it a "difference equation". The terms "difference equation" and "linear recursive relation" refer to essentially the same types of equations. Some people might reserve the term "difference equation" for an equation that is stated using the "difference operator" ##\triangle f(n) = f(n+1) - f(n) ## , but "difference equations" can be expressed as recursive relations.
 
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  • #5
Stephen Tashi said:
He make ##s## a function of ##a##.

He does not call the equation in question a "differential equation". He calls it a "difference equation". The terms "difference equation" and "linear recursive relation" refer to essentially the same types of equations. Some people might reserve the term "difference equation" for an equation that is stated using the "difference operator" ##\triangle f(n) = f(n+1) - f(n) ## , but "difference equations" can be expressed as recursive relations.
oh my error, this happen when you read fast i guess, I will continue studying this recursive relations they show up in many places
thanks for the help =D
 
  • #6
The recursive relation can be derived from the equation where you set a variable equal to the infinite sum of that variable's sequential derivatives: y = y' + y'' + y''' + ... + y^n'
Take the derivative of the above equation and subtract the derived equation from the first to obtain the recursive relation you stated: y = 2y'. Set y = c{n+1}(a) + c{n-1}(a), and y' then equals cos(a)*c{n}(a). I used the {} to indicate subscript because I'm a Luddite. The solution of the reduced form of the equation can be easily seen as: y=e^(x/2), or whatever variable you have in place of x as the independent, continuous parameter. Hope this helps! I've found that this infinite-order linear differential equation can be solved in this way by assuming that y^n' is equal to y^(n+1)' in the limit as n tends to infinity, due to the 1/2 in the exponent making both terms converge to zero in a manner similar to Cantor's theorem of fractals.
 

1. What is a recursive equation?

A recursive equation is an equation in which the solution for a particular term is dependent on the values of previous terms in the sequence. It is a type of mathematical function that is defined in terms of itself.

2. How are differential equations used to solve recursive equations?

Differential equations are used to solve recursive equations by modeling the relationship between the current term and the previous terms in the sequence. By finding the derivative of the recursive equation, a differential equation is created which can then be solved using various methods.

3. What is the difference between a linear and a nonlinear recursive equation?

A linear recursive equation is one in which the relationship between the current term and the previous terms is a linear function, while a nonlinear recursive equation is one in which the relationship is a nonlinear function. This means that the terms in a linear recursive equation are related by a constant rate of change, while in a nonlinear recursive equation, the rate of change varies.

4. What are some real-life applications of using differential equations to solve recursive equations?

Recursive equations and differential equations have many applications in various fields such as physics, economics, biology, and engineering. Some examples include modeling population growth, predicting stock market trends, and analyzing the behavior of electrical circuits.

5. What are the challenges of solving recursive equations using differential equations?

Solving recursive equations using differential equations can be challenging due to the complexity of the equations and the need for initial conditions to find a unique solution. Additionally, finding the appropriate differential equation to model a given recursive equation can be a difficult task.

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