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

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21 views

Recurrence relations and their solutions

I recently read an article about difference equations and found the solution of the fibonacci recurence there. It is this function: $f(n) = \frac{1}{\sqrt5}\left (\frac{1+\sqrt5}{2} \right )^{n}- ...
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0answers
19 views

Verifying solution of difference equation?

I have the following difference equation - $2h_{x+1} - 5h + 2h_{x-1} = 0$ for $x = 1, 2, ...., 19$ The boundary conditions are $h_0 = 1$ and $h_{20} = 0$ How would I go about verifying that $h_x = ...
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0answers
19 views

Difference Equations and their applications

What are some interesting applications to difference equations? I've learned about first and second order difference equations, first order systems, different kinds of equilibrium solutions (locally ...
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0answers
12 views

Elementary differential equations, difference equation

Find the effective annual yield of a bank account that pays interest at a rate of 7%, compounded daily; that is, divide the difference between the final and initial balances by the initial balance.
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1answer
27 views

Solving I. y[n+2]-(1/3)y[n+1]=sin(n) and II. y[n+2]+3y[n+1]-4y[n]=n-1 difference equations

I have two difference equations, which I just can't solve. I hardly even get the method, so if you could help me with the steps, I would be grateful. $y_{n+2}-\frac{1}{3}y_{n+1}=\sin(n)$ ...
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0answers
10 views

How to compute this Z-Transform?

The exercise is like this: $$y(k+1) - 3y(k) = 4^k$$ How do I compute $Z$ transform of $4^k$? I understand that I have to use the Z-Transform formula and the result after applying it is : $$sum ...
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2answers
22 views

How to analyze convergence of non-linear difference equation (recurrrence relations)

I've a couple of functions, such as: $Y(t+1)=2-\ln(Y(t))$ $Y(t+1)=(Y(t))^{-2}$ $Y(t+2)=e^{-Y(t)}$ and I need to analyze stability and convergence. No problem with stability, but I can't figure out ...
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0answers
10 views

Discrete Analoge Methods for solving difference equations

For solving non-linear first order differential equations we can use separation of variables (sometimes) or an integrating factor to convert a DE to an exact DE. Are there any analog methods for ...
2
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3answers
65 views

Why do we set $u_n=r^n$ to solve recurrence relations?

This is something I have never found a convincing answer to; maybe I haven't looked in the right places. When solving a linear difference equation we usually put $u_n=r^n$, solve the resulting ...
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1answer
39 views

Find the difference equation for {2, 4, 16, 256, …}

Write a difference equation to represent the change during the nth interval as a function of the previous term in the sequence. b. {2,4,16, 256,...} I know that an= 22n but I can't figure out how to ...
2
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0answers
24 views

Solution of partial difference equation

I want to find the explicit solution of the following difference equation $e_{i,j+1}=re_{i-1,j}+(1-2r)e_{i,j}+re_{i+1,j}+km_{i,j}$ where $r>0$, $k>0$ and $m_{i,j}$ are known and $e_{i,0}=0$. ...
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1answer
34 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
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1answer
65 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 ...
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2answers
126 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
58 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
65 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
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1answer
202 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
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1answer
72 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 ...
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0answers
14 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 ...
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2answers
48 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 ...
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0answers
27 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 ...
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1answer
62 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. ...
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2answers
37 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
19 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 $p_p$ ...
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1answer
56 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
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4answers
260 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 ...
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0answers
37 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 ...
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0answers
39 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? ...
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2answers
124 views

Using generation functions solve the following difference equation

Using generation functions solve the following difference equation $$ a_{n+1} - 3a_{n+2} + 2a_n = 7n ; n\geq0; a_0 = -1; a_1 = 3. $$
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1answer
32 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: ...
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5answers
316 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 ...
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3answers
63 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
56 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 ...
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1answer
28 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 $r$, ...
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1answer
24 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
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1answer
62 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 ...
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2answers
64 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 ...
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3answers
32 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??
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0answers
49 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
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0answers
52 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 ...
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1answer
63 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 ...
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2answers
81 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 ...
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3answers
118 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: $a_{n+1} = a_n - 0.069\cdot b_n$ $b_{n+1} = b_n + 0.069\cdot a_{n+1}$ ...
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0answers
36 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 $x[n]$, ...
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0answers
22 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 ...
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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
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2answers
133 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 ...
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1answer
90 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.
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1answer
145 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.
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31 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)}}= ...