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

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2answers
75 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
23 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
280 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
28 views

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

I am trying to solve the second order difference equation, ...
1
vote
2answers
44 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
26 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
votes
1answer
20 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
36 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
38 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
votes
0answers
37 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
41 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
58 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
vote
2answers
46 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
72 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
votes
0answers
27 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
vote
0answers
19 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
31 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
106 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
votes
1answer
49 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
65 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
27 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)}}= ...
-1
votes
2answers
82 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 ...
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0answers
60 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
23 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
votes
2answers
32 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 ...
1
vote
3answers
46 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
votes
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
votes
0answers
18 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
20 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
votes
0answers
18 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
61 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
votes
2answers
23 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 $\Delta_{t}{{r+t}\choose{t}}={{r+t}\choose{t+1}}$? ...
0
votes
4answers
56 views

How to solve this simple difference equation?

$ y(k+1)-2y(k)=k2^k$ I know that theres a formula for situations where the right hand side is a geometric series, but that doesn't seem to be the case.
0
votes
1answer
23 views

Rate and time calculation difference

A person is running at 8.5 mph and sprints at 14.5 mph. How much of a time difference will there be if he starts sprinting with 1/10 mile left and with 30 seconds (sprinting) left? Why?
1
vote
0answers
19 views

An expression for 2-dimensional element as a sum of differences of elements?

For 1-dimension, it's simple. $a_{n}=a_{1}+\sum_{i=2}^{n}(a_{i}-a_{i-1})$ But what would be the corresponding identity for 2-dimension? In other words, if we put $a_{n,m}=a_{1,1}+X$ then how can ...
1
vote
0answers
43 views

“Strange” plot of a difference equation

In this book about intertemporal optimization (page 33) I've found this difference equation: $x_{t+1}=ax_t \quad, \quad a>0$ The solution is: $x_t=a^t x_0$ where $x_0$ is the initial value of ...
0
votes
1answer
19 views

Difference $\Delta P_t$ approaches 0, then its relative difference $\Delta P_t / P_(t-1) \approx ln(P_t / P_(t-1))$.

When difference $\Delta P_t$ approaches 0, its relative difference $\dfrac{\Delta P_t}{P_{t-1}} \approx \ln(\dfrac{P_t}{P_{t-1}})$. I know that it can be shown somehow with Taylor series: ...
0
votes
1answer
37 views

Computation of $n$-th order difference of falling factorial

I was reading a difference equation textbook and came across a problem. The question asks to compute ${\Delta}^nt^{\underline3}$ for $n=1,2,3,...$, where $t^{\underline3}$ is the falling factorial ...
2
votes
1answer
32 views

Difference equation: $y_{k+1} = y_{k} + \frac{c}{2k}$

I'd like to solve this difference equation. Unfortunately, the forcing term is not geometric, so I don't know how to find the solution: $$ y_{k+1} = y_k + \frac{c}{2k}. $$
0
votes
1answer
184 views

What values of $p$ make this a transient chain?

Suppose we have a Markov chain with state space $S = \{0, 1, 2, \dots \}$ and probabilities $p(x, x + 2) = p$, $p(x, x - 1) = 1-p$ for $x > 0$ and $p(0, 2) = p$ and $p(0, 0) = 1 - p$ I would ...
2
votes
1answer
52 views

Nonlinear difference equation

Maybe this is a trivial question, but how to find the general solution to the following first order difference equation? $$ y_{t+1}=a+\frac{b}{y_{t}} $$ Also, could someone recommend a reference ...
2
votes
1answer
110 views

Numerical Solution of difference equation

I am trying to solve a nonlinear difference equation of the form: $x_{i+1} = f(x_i, x_{i-1})$ for $i = 0,\ldots,N-1$ with given boundary conditions $x_0 = a$ and $x_N = g(x_{N-1})$ where $f$ and $g$ ...
0
votes
0answers
71 views

Computing the steady state probability vector of a random walk on $\{0, 1, \dots, n\}$

Suppose we have a random walk on $\{0, \dots, n\}$ with transition probabilities $$P(x, x + 1) = p \\ P(x, x - 1) = 1 - p$$ for $1 \le x \le n -1$, $P(0, 1) = a$, $P(0, 0) = 1 - a$, $P(n, n - 1) = ...
1
vote
1answer
54 views

How to find solutions to the differential equation created by setting the Schwarzian Derivative equal to zero?

I'm in a Masters level course on difference equations, and last week we discussed theorems which can be applied to show the stability characteristics of non-hyperbolic equilibria of first order ...
0
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0answers
65 views

Catalan numbers via partial difference equations?

It is known that Catalan numbers can be characterized in the following way: let $f(n,k)$ be a function of two integer variables, such that the following recurrence holds: $$f(n+1,k+1) = f(n,k) + ...
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0answers
32 views

Analyzing convergence of a simple difference equation

I was playing with my calculator, had an arbitrary number on the screen, then pressed 1/x ,then sqrt, then 1/x, and so forth. I noticed it converged to 1.0 after a few back-and-forths. So, it occurred ...
0
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1answer
46 views

Show that this is a general solution of the difference equation

I am currently doing my homework and have been struggling to pass this question: The difference equation Un = Un-1 + Un+1 is a discrete model for the equilibrium heat distribution along a straight ...
0
votes
1answer
24 views

Stability of difference equation considering only positive values

I'm analyzing the stability of such system difference equation with the constraint that $y_n \geq 0$ $\forall n \geq 0$ : $y_n = B y_{n-1} + D y_{n-2} \enspace (1)$ Using variable transform, the ...
0
votes
0answers
93 views

Sum of two Bessel function of first kind

I want to find an expression for the sum of two Bessel functions of first kind with the same argument but a different order, i.e. $F(i,j)=|J_{i+j}(x)+(-1)^j J_{i-j}(x)|^2$. Is there any way of ...