A nonlinear difference equation

In summary, the conversation discusses an equation and a special solution that can be used to solve it. The equation is derived from trigonometry and can be solved using trig substitution. Mathematica provides a general solution, but an analytic method for finding a special solution is not mentioned.
  • #1
asmani
105
0
Consider the equation
[tex]\frac{a_n-a_{n-1}}{1+a_na_{n-1}}=\frac{1}{2n^2}[/tex]
I know one special solution is
[tex]a_n=\frac{n}{n+1}[/tex]
But how to solve and find the general solution?

Thanks in advance.
 
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  • #2
hi asmani! :smile:

hint: trig substitution :wink:
 
  • #3
Thank you tiny-tim. I don't think that works, since the the equation is actually coming from trigonometry!

Mathematica gives the following solution:

[tex]a_n=\frac{a_0+\left (1+a_0 \right )n}{1+\left ( 1-a_0 \right )n}[/tex]

But I don't know how to derive this solution analytically.

P.S. The original problem was to show that:

[tex]\sum_{n=1}^{\infty}\tan^{-1}\left (\frac{1}{2n^2} \right )=\frac{\pi}{4}[/tex]
 
  • #4
asmani said:
Thank you tiny-tim. I don't think that works …

yes it does!

try it! :smile:
 
  • #5
[itex]a_n=\tan\theta_n [/itex]? Another hint please!
 
  • #6
asmani said:
[itex]a_n=\tan\theta_n [/itex]?

yes! :biggrin:

so the LHS of the original equation is … ? :smile:
 
  • #7
I think I get it, Thanks a lot.

[tex]\tan \left (\theta_n-\theta_{n-1} \right )=\frac{1}{2n^2}[/tex]
and then
[tex]\theta_n =\tan^{-1}\left (\frac{1}{2n^2} \right )+\theta_{n-1}[/tex]
and then
[tex]\theta_n =\theta_{0}+\sum_{k=1}^{n}\tan^{-1}\left (\frac{1}{2k^2} \right )[/tex]
By knowing that special solution mentioned in post #1, I can derive the formula in post #3. What If I didn't know that special solution?
 
Last edited:
  • #8
hmm :confused:

putting ao = 0 or 1 gives an = 1 + 2n or -1/(1 + 2n)

and putting ao = ±i looks interesting :rolleyes: (i haven't followed it through :redface:) …

do either of those help? :smile:
 
  • #9
Putting a0=0 in which equation?
 
  • #10
Mathematica's …
[tex]a_n=\frac{a_0+\left (1+a_0 \right )n}{1+\left ( 1-a_0 \right )n}[/tex]
 
  • #11
OK. So far, first I 'guessed' a special solution, then I derived the general solution by using that special solution. Is there any analytic way to find (not guess) a special solution?
 

1. What is a nonlinear difference equation?

A nonlinear difference equation is a mathematical equation that describes the relationship between a variable and its previous values, where the relationship between the current and previous values is not a simple linear function. This means that the change in the variable from one time step to the next is not directly proportional to the current value.

2. How is a nonlinear difference equation different from a linear difference equation?

A linear difference equation has a relationship between the current value and its previous value that can be described by a linear function, such as y = mx + b. This means that the change in the variable is directly proportional to the current value. In contrast, a nonlinear difference equation has a relationship that cannot be described by a linear function, and the change in the variable is not directly proportional to the current value.

3. What are some real-world applications of nonlinear difference equations?

Nonlinear difference equations are commonly used in fields such as physics, biology, economics, and engineering to model complex systems that exhibit nonlinear behavior. Some examples include population dynamics, chemical reactions, weather patterns, and stock market fluctuations.

4. How are nonlinear difference equations solved?

Solving a nonlinear difference equation can be a complex task and often requires the use of numerical methods or computer simulations. In some cases, an analytical solution may be possible, but this is not always the case. It is important to carefully consider the specific equation and its parameters before attempting to solve it.

5. What are some challenges associated with studying nonlinear difference equations?

One of the main challenges of studying nonlinear difference equations is that they often exhibit chaotic behavior, making it difficult to predict their long-term behavior. Additionally, the complexity of these equations can make them difficult to solve analytically, requiring the use of numerical methods or computer simulations. It is also important to carefully consider the parameters and initial conditions in order to accurately model the system.

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