4
votes
0answers
42 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 ...
4
votes
4answers
257 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 ...
-2
votes
2answers
107 views

Using generation functions solve the following difference equation

Using generation functions solve the following difference equation $$ a_{n+1} - 3a_{n+2} + 2a_n = 7n ; n\geq0; a_0 = -1; a_1 = 3. $$
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 ...
1
vote
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
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 ...
1
vote
2answers
61 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
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 ...
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 ...
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: $a_{n+1} = a_n - 0.069\cdot b_n$ $b_{n+1} = b_n + 0.069\cdot a_{n+1}$ ...
1
vote
0answers
79 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
votes
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 ...
2
votes
1answer
64 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
129 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
138 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 ...
1
vote
2answers
34 views

Fixed points for $k_{t+1}=\sqrt{k_t}-\frac{k_t}{2}$

For the difference equation $k_{t+1}=\sqrt{k_t}-\frac{k_t}{2}$ one has to find all "fixed points" and determine whether they are locally or globally asymptotically stable. Now I'm not quite sure ...
2
votes
0answers
51 views

Finding the best real value for $C$.

Consider the recurrence $f_{n+1}=f_n + \ln(f_n)$ with $f_0=2$. Also consider differential equations of type $g(0)=2$ and $\dfrac{d g}{d x}=\ln(g(x)- C \cdot \ln(g(x)))$. Lets call the solution ...
0
votes
1answer
207 views

recurrence relation of integral

Consider the integral defined by $$\displaystyle{ I_k( \phi) = \int_0^{\pi} \frac{ \cos(k\theta) - \cos( k \phi) }{ \cos \theta - \cos\phi} d \theta} $$ (a) Show that $I_k( \phi) $ satisfies the ...
3
votes
2answers
165 views

Given $g(x)$, how to solve function recurrence $f(x)=af(\alpha x)+bf(\beta x)+g(x)$ where $\alpha\neq\beta$

If we have a recurrence like $$f(x)=af(\alpha x)+bf(\beta x)+g(x)$$ where $a,b,\alpha,\beta\in\mathbb{R}$ and $\alpha\neq\beta$ and $g(x)$ is known. How can we solve this kind of recurrence? For ...
3
votes
1answer
136 views

A difference equation related to RW on Hypercube

I am trying to solve the following recurrent relation $$ T(n)=\frac{n}{m}T(n-1)+\frac{m-n}{m}T(n+1)+1, \,\, \text{subject to } T(m)=0 $$ Where $0\leq n\leq m$ and $m$ is a fixed integer. I have ...
1
vote
4answers
160 views

Reccurence relation: Lucas sequence

I need to solve the given recurrence relation:$$L_n = L_{n-1} + L_{n-2},$$ $n\geq3$ and $ L_1 = 1, L_2 =3$ I'm confused as to what $n\geq3$ is doing there, since $L_1$ and $L_2$ are given I got ...
4
votes
4answers
1k views

Closed form solution of Fibonacci-like sequence

Could someone please tell me the closed form solution of the equation below. $$F(n) = 2F(n-1) + 2F(n-2)$$ $$F(1) = 1$$ $$F(2) = 3$$ Is there any way it can be easily deduced if the closed form ...
1
vote
1answer
158 views

How to solve a recurrence equation with non-constant coefficients?

How to solve a recurrence equation with non-constant coefficients? The equation is $$ 120(3k+1)(18k^3-21k-2)(k+2)a_{3k-6}=120(54k^4-117k^2+4)(k-1)a_{3k-5}+(k-1)^2(3k^2-2)^2k^3(k+2)(k+1)(6k^2+6k-1). $$ ...
0
votes
0answers
60 views

properties about number of groups of linearly independent solutions in the general solutions of linear functional-differential equations

Are there any effective methods to determinate the number of groups of linearly independent solutions in the general solutions of linear functional-differential equations of the form ...
1
vote
3answers
286 views

general solution to linear second order difference equation

Is there a general solution to difference equations of the form: $$ u(n+2) + a(n)u(n+1) + b(n)u(n) = 0 $$ Thank you in advance
1
vote
3answers
135 views

How do you go about solving difference equations?

Say you have something of the form $p_1 = p$ $p_n=kp_{n-1}+(1-k)(1-p_{n-1})$ How does one go about finding $p_{n}$ in terms of $n,p$ and $k$? In my notes here's how it's found $p_n-1/2 = ...
7
votes
5answers
456 views

How does one find a formula for the recurrence relation $a_{1}=1,a_{2}=3, a_{n+2}=a_{n+1}+a_{n}?$

I am self-studying Discrete Mathematics, and I have two exercises to solve. Find a formula for the following recurrence relation: (translated from Portuguese) a) $a_{1}=3,a_{2}=5, ...
0
votes
3answers
496 views

can we use generating functions to solve the recurrence relation $a_n = a_{n-1} + a_{n-2}$, $a_1=1$, $a_2=2$?

I have this question. Can we use generating functions to solve the recurrence relation $$\begin{align*} a_1 &= 1,\\ a_2 &= 2,\\ a_n &= a_{n-1} + a_{n-2} \end{align*}$$ Thanks
3
votes
2answers
386 views

Particular solution of recurrence equations

How do we solve recurrence equations of the form: $$ax_{n+1}+bx_n+cx_{n-1}=dn^p+e\;,$$ where $a,b,c,d,e$ are constants and $p\in \mathbb Z$? Perhaps we could first solve the homogeneous equation ...
7
votes
8answers
830 views

Need help deriving recurrence relation for even-valued Fibonacci numbers.

That would be every third Fibonacci number, e.g. $0, 2, 8, 34, 144, 610, 2584, 10946,...$ Empirically one can check that: $a(n) = 4a(n-1) + a(n-2)$ where $a(-1) = 2, a(0) = 0$. If $f(n)$ is ...
1
vote
1answer
101 views

Recovering two first terms from sequence $f(n)=f(n-1)+f(n-2)$

Having very simple sequence $f(n)=f(n-1)+f(n-2)$ and having $n-th$ term given how can we calculate from which first two terms this $n-th$ term came? I know the answer can be not unique so highest ...
2
votes
2answers
2k views

Characteristic equation of a recurrence relation

Yesterday there was a question here based on solving a recurrence relation and then when I tried to google for methods of solving recurrence relations, I found this, which gave different ways of ...
2
votes
2answers
168 views

Recurrence $A(n,x)=p(x)A(n-1,x-1)+q(x)A(n-1,x)$

What is the general approach to solving this recurrent equation given that $p(x)$ and $q(x)$ are not constant and do not depend on $n$ and $p(x)+q(x) \neq 1$. Please just give me some hints, don't ...
0
votes
1answer
86 views

Can we express $p_n$ in terms of $p_0, p_1$ and $n$?

$p_0=a$, $p_1=b$, $bp_n=p_{n+1}+p_{n-1}$ express $p_n$ in terms of $a,b,n$. Any help would be appreciated, because you guys are splendid.
4
votes
1answer
222 views

What types of functions do recurrence relations methods apply to?

I have been working with a function that I defined recursively as $$a(n) = (1-a(n-1)^k)^k$$ where $a(0) = x$ and $k$ is an integer $>1$. So really, $a(n)$ returns a function on $x$ and $k$. I have ...
2
votes
1answer
456 views

Correspondence between ODE and difference equation

In Wikipedia about difference equations, there is some description about correspondence between ODE and difference equation: If you consider the Taylor series of the solution to a linear ...