Questions related to difference equations, which are discrete analogs of differential equations.

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0
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1answer
31 views

$\Delta^kn^\alpha$ converges monotonically to zero when $\alpha<k$

I am trying to prove that the $k$-th finite difference of the series $n^\alpha$ converges to zero monotonically as $n\to\infty$ when $\alpha<k$. The differential analogue is ...
2
votes
1answer
54 views

Limit of a Discrete Dynamical System

For the system defined below, the point by point evolution remains bounded for all $t$ so I could see that some sort of limit exists. However, the question is what sort of limit is it -- a single ...
0
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2answers
122 views

Why do we use the term “equivalent” with Operators but “equal” with Functions?

Why do we speak in terms of "equality" when we deal with functions but "equivalence" when dealing with operators? To elaborate: Two functions, f and g are equal to each other (denoted: f=g) if: ...
3
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0answers
54 views

Boundedness of solutions of Difference equation

Consider a second order difference equation in complex plane, \begin{equation} z_{n+1}=\frac{\alpha + \beta z_{n}}{1+z_{n-1}},\qquad n=0,1,\ldots \end{equation} where the parameters $\alpha, ~\beta$ ...
4
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0answers
41 views

How to solve a non-homogeneous second-order linear difference equation with both a forward and a backward difference?

Quite a long title for this: I'm looking for the general solution of the following difference equation: $$ax_{t+1} -bx_t + x_{t-1} = c + u_t$$ where $a,b,c$ are real constants and $u_t$ is a bounded ...
3
votes
1answer
195 views

How to solve this recurrence Relation - Varying Coefficient

Sir,I have two questions related to this recurrence relation. It has been messing with me for long. Because of this I couldn't proceed my work for some time .This contains a polynomial term n+2 in ...
2
votes
1answer
70 views

a system of finite difference equations

Let $a,b>0$ such that $ab<1$ consider the system$$x_{t+1}=x_ty_t+ay_t$$ $$y_{t+1}=x_ty_t+bx_t$$ I would like you to help me answer the following: find values $a$ and $b$ ​​for which the ...
0
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0answers
11 views

Literature on functional difference equations

dear community. I'm looking for books/guides on functional difference equations. Can you recommend some? Below I try to explain what kind of equations I have in mind. As an example, one of the ...
0
votes
2answers
36 views

Calculating a percentage between two numbers

I have two numbers, a minimum value, and a maximum value. I also have a percent. This percent helps me find a value between the two numbers, the minimum value and the maximum value. I cannot figure ...
1
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0answers
25 views

Is there a better notation for difference equations?

Difference equations are quite messy to deal with, esp. in constraint optimization with many time subscripts that invite mistakes. Is there a better notation? Something like Feynman diagrams for ...
0
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1answer
42 views

General solution of a system of linear differential equations with multiple generalized eigenvectors

I am looking for general solutions for the linear sODE's $$\textbf{x}'(t) = A\textbf{x}(t)$$ with $t \geq 0$ and $A \in \mathbb{R}^{n \times n}$ Let focus on just real eigenvalues and eigenvectors. ...
0
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2answers
34 views

Is there a difference for discount per unit and discount per purchase total?

I can't find relevant tags for my question so I wonder if this is a good place to ask. I wanted to ask this a long time ago but keep forgetting. Let's suppose when shopping for 3 units of specific ...
0
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0answers
18 views

difference equation soling

Need to help sovling a differene quation :) $p_t = - \frac{p_{t-1} + \alpha + \gamma \beta}{\delta \beta}$ My Thoughts: $p_t = p_{CF} + p_p$ where $p_{CF}$ is the complementary function and $p_p$ ...
1
vote
1answer
53 views

Example for finite dimensional analog of integral transforms

I understand that integral transforms are generalisations of the dot product of functions that could be interpreted as infinite dimensional vectors. The most significant advantage then is that ...
4
votes
4answers
256 views

Finding the billionth number in the series: $2, 3, 4, 6, 9, 13, 19, 28, 42, \ldots $?

Series is defined as $$a_{n+1} = \lfloor\frac{3\cdot a_n}{2}\rfloor,\qquad a_0 = 2$$ It can be viewed as the number of animals starting from a single pair if any pair of animals can produce a single ...
1
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0answers
36 views

$\cos(2\arccos(\frac{a}{a+1})x$

I have trying to prove that this cosine map: $$\frac{r}{4}((a+1)\cos\left(2\arccos\left(\frac{a}{a+1}\right)\ \left(X_n-\frac12\right)-a\right)$$ is a logistic map. What I have done so far: Using ...
1
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0answers
35 views

Closed form for a sequence defined recursively

Let $a_k$ be a sequence such that $a_0=0, a_1=0, a_2=1, a_3=1$ and $$a_{k+4}=-\frac{a_{k}+ka_{k+2}}{(k+1)(k+2)}$$ for $k\ge 0$. My question is: Is a closed form formula for $a_n, n\ge 4$ possible? ...
-2
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2answers
105 views

Using generation functions solve the following difference equation

Using generation functions solve the following difference equation $$ a_{n+1} - 3a_{n+2} + 2a_n = 7n ; n\geq0; a_0 = -1; a_1 = 3. $$
0
votes
1answer
29 views

Linear Difference Equations and how to solve for $y_n$

I am currently trying to study difference equations for my first year undergrad Calculus course. I am struggling to understand how they work. I am currently trying this question: ...
5
votes
5answers
312 views

What particular solution should I guess for this relation?

Just trying to solve a non-homogeneous recurrence relation: $$f(n) = 2f(n-1) + n2^n$$ $$f(0) = 3$$ Characteristic equation is: $$f(n) - 2f(n-1) = 0$$ $$a-2 = 0$$ $$a = 2$$ Homogeneous ...
0
votes
3answers
56 views

How to find the particular solution of a second order difference equation

I am trying to solve the second order difference equation, ...
1
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2answers
52 views

difference equation( recurrence relation)

Let $y_n$ satisfy the nonlinear difference equation: $$(n+1)y_n=(2n)y_{n-1}+n.$$ Let $u_n=(n+1) y_n$. Show that $$u_n= 2u_{n-1}+n.$$ Solve the linear difference equation for $u_n$. Hence find ...
0
votes
1answer
27 views

second order difference equation question

$y_{n+2} - 2y_{n+1} + 2y_n = 62^n$ sub $y_n= r^n$ then $y_{n+2}=r^{n+2}$, $y_{n+1}=r^{n+1}$ so $r^{n+2} - 2r^{n+1} + 2r^n = 0$ $r^n( r^2 - 2r + 2) = 0$ I got a problem here, I can solve for $r$, ...
0
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1answer
22 views

Help with difference/recursion equation change of variable

I am in a self study of Dynamic Systems and am reading through David Luenberger's book and cannot seem to figure the following question out. Solve the difference equation using a change of variables ...
2
votes
1answer
47 views

second-order difference equation

I have a second-order difference equation question. yn + 2 - 78yn = 23n^2 What is the value of root in auxiliary equation? I have tried searching for videos online but I don't really quite ...
1
vote
2answers
60 views

Reference Request: Difference Equations

I am taking a second course in calculus and came across sequences defined inductively, as in recursively. My teacher told the class that a general formula for the $n$th term can be obtained using a ...
0
votes
3answers
31 views

Is this possible? - Controlled system equation

Does this mathematical equation even make sense. This is taken From my controlled system book, where 4.12 doesn't make sense for.. How come is that true??
0
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0answers
45 views

Linear Constant Coefficient Different Equation

The question I have is about linear constant coefficient question but I don't really know for sure how to do it. The question is: Suppose that $N_{m+1}-N_m=f(N_m,N_{m-1})$.(a) How would you determine ...
3
votes
0answers
46 views

Conditions of a Monotonic Process?

$f$ is the output of a discrete time process described by $f(k)=\sum_{i=1}^{k-1}w_{ki}f(i)$ where $f(1)\geq0$ is a known initial condition and $w_{ki}\geq0$ are weights of previous states on the ...
1
vote
1answer
62 views

forming difference equation

there is a square with $60$ equal blocks. If a mosquito(bug)is set to fly starting at block $1$, it is equally likely to fly to other blocks. what is the probability after $n$ flies, the mosquito is ...
1
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2answers
65 views

Is there a method for finding the fixed point of logarithmic functions?

I am faced with this function (warning, I am not good at math) $x(t+1)=0.5 \ln x(t)+1$ initial condition = 1 . I know the fixed point is 1 because $0.5 \ln (1)+1=1$ but I wanted to know the ...
3
votes
3answers
104 views

Why does this recurrence relation generate a sinusoidal curve?

I came across the following coupled recurrence relation while watching this video called Media for Thinking the Unthinkable: $a_{n+1} = a_n - 0.069\cdot b_n$ $b_{n+1} = b_n + 0.069\cdot a_{n+1}$ ...
0
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0answers
32 views

Sufficient condition for shift-invariance of linear constant coefficient difference equation

What is the sufficient condition for a LCCDE, defined by $$\sum_{k=0}^{N}a_{k}y[n-k] = \sum_{k=0}^{M}b_{k}x[n-k]$$ to be shift-invariant? (shift-invariance: if $y[n]$ is the solution for input $x[n]$, ...
1
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0answers
21 views

Difference equation - counting problem

I need to to define difference equation for following problem and solve that equation using generating function. Border of length 10cm is made of small bricks (10cm long) and large bricks (20cm ...
1
vote
1answer
33 views

Difference equation formula $\sum a^t = \frac{a^t}{a-1}$.

As I explain below, this question was originally posted by user YYG, but then deleted. I am reposting the question (from memory) and I will answer it myself below. Question: In Difference Equations ...
3
votes
2answers
129 views

A proof using $\Delta^ny(t)=\sum_{k=0}^{n}{(-1)^k\dbinom{n}{k}y(t+n-k)}$

Please How can I use $\Delta^ny(t)=\sum_{k=0}^{n}{(-1)^k\dbinom{n}{k}y(t+n-k)}$ to prove $\sum_{i=0}^{n}{(-1)^i\dbinom{n}{i}y(i)}=(-1)^n\Delta^ny(0)$ and hence to evaluate ...
0
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1answer
78 views

Summation of falling factorials

I just want to know if I should evaluate $\sum(t+1)^\underline{4}$ the way we evaluate $\sum{t^\underline{4}}$. Thanks.
1
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1answer
101 views

Negative falling Factorial

Please can someone tell me what is the value of $1^\underline{-2}$? I know that $1^\underline{2}=0$. Thanks.
0
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0answers
29 views

Questions on Difference operators

Please I really need help on the following short problems on difference operators that I need even some clues on how to go by them: 1) $\sum_{t=1}^{4}{\dfrac{1}{(t+1)(t+2)(t+3)}}= ...
0
votes
2answers
98 views

Prove (1 + x)^n + (1 - x)^n < 2^n by using Binomial Theorem

Hi my boss asked me to resolve this equation: Prove (1 + x)^n + (1 - x)^n < 2^n by using Binomial Theorem -1 < x < 1 ...
1
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0answers
77 views

What is common between difference operators and recurent relations?

They say that solutions of recurrence relations are combinations of exponential functions, the series like [1 a a^2 a^3 and etc]. I know that the difference operators have a matrix like ...
1
vote
1answer
24 views

2nd order homogeneous repeated roots

$$a_{n+2}-6a_{n+1}+9a_{n}= 3^n \quad n\geq 0 \quad a_{0}=2 \quad a_{1}=3$$ Got repeated roots of 3, so $a_{n}= A\cdot3^n+b\cdot (n\cdot3^n)$ how would i calculate A+B when there is $3^n$? Edit: So ...
0
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2answers
35 views

2nd order homogeneous difference equation

$$a_{n+2} = 9a_{n+1} - 18a_n,\quad n\geq 0,\,\,a_0=1,\,\, a_1=3$$ I got to the point where i moved all to LHS which gives me $a_{n+2} - 9 a_{n+1} + 18 a_n$ (correct me if I'm wrong). I then ...
0
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3answers
47 views

Sequence difference equatiom

For $n \ge 2$ the terms in the sequence $a = \{1, 6, 17, 45, 118, 309, \ldots\}$ are related by the difference equation $$a_{n+2} = \boxed{\phantom{XX}} \, a_{n+1} + \boxed{\phantom{XX}} \, a_n $$ ...
0
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0answers
31 views

A second-order difference equation

Fix a positive integer $r\geq 2$. For each integer $k\geq 2$, we have the recursion $$ \left(k + 2\right)\left(k + \frac{1}{3}\right)a_{k} - 2\left(k + 1\right) \left(k - 1 + r\right)a_{k-1} + \left(k ...
0
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0answers
23 views

Theorem of finding stability of equilibrium points in a difference equation

We have been given a theorem on equilibrium points of difference equations which say: if $x(n+1) = f(x(n))$ and $f'(x^*) = 1$ then: i) $f''(x^*) \neq 0$ then $x^*$ is unstable ii) if $f''(x^*) = ...
0
votes
1answer
25 views

What is the steady state for this difference equation: $X_{n+1}-X_{n}+\beta \alpha X_{n-1}(1-\frac {X_{n-1}}{X_{max}})=t$

This is my self study, as I know the steady state from an difference equation should satisfy $x=X_{n+1}=X_{n}$ What is the steady state for this difference equation? $$X_{n+1}-X_{n}+\beta ...
0
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0answers
20 views

A word problem in difference equations

The following problem is from Difference Equations by W. G. Kelly and A. C. Peterson (2nd edition). I just couldn't figure out where to start. 'Suppose that $t$ points are chosen on the perimeter of ...
0
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0answers
67 views

Solving a log-linear equation forward.

Hopefully someone can help me with this problem. I have gotten a bit stuck when trying to solve the following log-linearized equation forward and to obtain a fundamental soulution. My equation is of ...
0
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2answers
29 views

Difference operation on factorials

Please how is the combination addition formula ${{t}\choose{r}}={{t-1}\choose{r}}+{{t-1}\choose{r-1}}$ useful in proving the difference equation $\Delta_{t}{{r+t}\choose{t}}={{r+t}\choose{t+1}}$? ...