What is Difference equation: Definition and 97 Discussions
In mathematics, a recurrence relation is an equation that recursively defines a sequence or multidimensional array of values, once one or more initial terms are given; each further term of the sequence or array is defined as a function of the preceding terms.
The term difference equation sometimes (and for the purposes of this article) refers to a specific type of recurrence relation. However, "difference equation" is frequently used to refer to any recurrence relation.
I'm studying Differential Equations from Tenenbaum's, and currently going through non-homogeneous second order linear differential equations with constant coefficients. Method of Undetermined Coefficients is the concerned topic here. I will put forth my doubt through an example. Let's say we are...
We know
$$
K(x,t) = \frac{1}{\sqrt{4\pi t}}\exp(-\frac{x^2}{4t})
$$
is a solution to the heat equation:
$$
\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}
$$
I would like to ask how to prove:
$$
u(x,t) = \int_{-\infty}^{\infty} K(x-y,t)f(y)dy
$$
is also the solution to...
I want to solve this using difference equation. So I set up the general equation to be
Pi = 0.5 Pi+1 + 0.5 i-1
I changed it to euler's form pi = z
0.5z2-z+0.5 = 0
z = 1
since z is a repeated real root
I set up general formula
Pn = A(1)n+B(1)n
then
P0 = A = 1
PN = A+BN = 0 -> A= -BN...
Hi hi, I'm confused about how to mix this two concepts, actually the wave equation:
##\frac {\partial^2 u} {\partial t^2} = v_x^2 \frac {\partial^2 u} {\partial x^2} + v_y^2\frac {\partial^2 u} {\partial y^2} + force##
The equation will apply the rule all over the space, but I have the next...
Hi,
I am trying to work out how I could obtain an expression for a z-transform for the following expression:
x_{n + 1} = r x_{n} \left( 1 - x_{n} \right)
I am hoping to derive X(z) and then use the final value theorem to show agreement with numerically calculated steady state values.
I...
How do I start this? I plugged the differential equation at wolfram alpha and it semmed too complicated for such an exercise. I've also looked at a sample of an answer on cheeg where the initial approach is to rewrite the equation as ##\frac{d}{dt} (\frac{\dot\theta^2}{2}-cos(\theta)) = 0##
How...
Homework Statement
Hi, I'm trying to calculate the formula for the position vs. time of a rocket landing from an altitude of 100km. I'm neglecting a lot of forces for simplification but basically, I want to solve ##F_{net} = Drag - mg##.
Homework Equations
Drag Force: D = ## \frac {C_dAρv^2}...
Hi all,
I am a bit new in this, am trying to learn DE, dynamical systems, & chaos. I am looking into some answers for the following questions:
1) Is it always possible to derive a difference equation for every differential equation, and if so how do we do that?
2) Consider Lorenz system...
Hello,
If I have a homgeneous linear differential equation like this one (or any other eq):
$$y''(x)-y'(x)=0$$
And they give me these Dirichlet boundary conditions:
$$y(0)=y(1)=0$$
Can I transform them into a mixed boundary conditions?:
$$y(0)=y'(1)=0$$
I tried solving the equation, derivating...
Hi, I am working on a project where the normal process has been to push a single linear plain away from a given area. I am now considering lifting the single linear plain away but wondered about the physical dynamics when comparing pushing to lifting something.
In the image:-
A. Represents...
Homework Statement
Finding the general solution:
y”+4y’+4y=t*e^(-2t)
Homework EquationsThe Attempt at a Solution
So I got the complementary solution pretty easily as y= c1*e^(-2t)+c2*te^(-2t)
I haven’t been able to find a particular solution using the method of undetermined coefficients. I...
Hi, I'm a high school graduate who's been going through some UNISA (University of South Africa) mathematical modelling material and I've come across the following difference equation -
a(n+1) = 1 - a(n).
The answer given in the material was that the solution alternates between a(0) for n even...
In the following question I need to find the Fourier cosine series of the triangular wave formed by extending the function f(x) as a periodic function of period 2
$$f(x) = \begin{cases}
1+x,& -1\leq x \leq 0\\
1-x, & 0\leq x \leq 1\\\end{cases}$$
I just have a few questions then I will be able...
I have a little question. I want to know if there is a process in which I can find equilibrium solutions to some system of difference equations. For example, if I have something crazy like
$$\begin{cases} x[n+1]=(x[n])^2y[n]+z[n]e^{-ax[n]} \\
y[n+1]= z[n]x[n]+x[n+1]y[n+1]\\
z[n+1]=...
Homework Statement
A block is attached to the sides of a square box by 4 springs. The box is placed horizontally on a frictionless surface (ignore gravity). The mass of the block is ##m##, the natural length of each spring is ##l##, and the strength of each spring is ##k##. Place the block at...
Homework Statement
A block is attached to the sides of a square box by 4 springs. The box is placed horizontally on a frictionless surface (ignore gravity). The mass of the block is ##m##, the natural length of each spring is ##l##, and the strength of each spring is ##k##. Place the block at...
Can anyone help me solve this? Text goes: solve the difference equation first directly, then with generating functions
I've been stuck with it for hours. I have no idea what to do with "2n+3". We don't have anything about this special case in my textbook, and I can't seem to find anything...
Homework Statement
My question: Can I turn this difference equation for R below, into a continuous function R(t)? I have no idea if, or how, I can. And I'd like to.
Equation derived from the following manufacturer statement on the thermal response of a thermistor to a fixed temperature:
The...
The Complete Idiot's Guide to Calculus
INTRODUCTION
I've never really been very good at math and when I found out I had to take a Calculus class I started to panic. Once I gathered myself I went to the local bookstore to see if I could get a book to read so i could get a heads start. We are...
Homework Statement
\begin{equation}
(1-x)y^{"}+y = 0
\end{equation}
I am here but do not understand how to combine the two summations:
Mod note: Fixed LaTeX in following equation.
$$(1-x)\sum_{n=0}^{\infty}(n+2)(n+1)a_{n+2}x^n+\sum_{n=0}^{\infty}a_nx^n = 0$$
A cone-shaped water tank is given by V(h)=\pi(h-\frac{h^2}{3}+\frac{h^3}{27})
Show using Torricelli's law law that
-2\sqrt2\pi(\frac{1}{\sqrt{h}}-\frac{2}{3}\sqrt h+\frac{1}{9}h^{3/2})\frac{dh}{dt}=1
What I have done so far:
V'=\frac{1}{9}\pi(h-3)^2h'
Homework Statement
When drugs are used to treat a medical condition, doctors often recommend starting with a higher dose on the first day than on subsequent days. In this problem, we consider a simple model to understand why. Assume that the human body is a tank of blood and that drugs...
Homework Statement
Homework Equations
The equation describing the balance will be f(n+1)=f(n)+R/12*Dm-Cf
with f(n)=initial deposit
R=Annual Rate
Dm=Each mouth Deposit 150
Cf= each month fee
The Attempt at a Solution
Can someone shed some lights on it?
Thanks[/B]
Hello!
I started to study a differential equation, and turned the problem into a difference equation (a_0, a_1 \in \mathbb{R}, a_2 = 0, a_3 = \frac{a_0 + a_1}{6})
a_{v+2} = \frac{v^2a_v + a_{v-1}}{(v+1)(v+2)}, where v \ge 2.
The numbers a_v are coefficients of a serie solution to the original...
I'm pretty rusty at calculus. I did well in them, but my memory is terrible and I have forgotten a lot.
I'm going to take ordinary differential equations (it looks and sounds like an intro DE class with some linear algebra too) next spring. What should I study and what not to prepare for this...
I have a difference equation which is given as:
ΔP = e^P [1]
where we can re-write ΔP as: Δ P = P_2 - P_1, where the subscripts indicate two distinct discrete time indices.
What I would like to do: is to convert this into a continuous time expression and solve it, if possible.
In order...
Homework Statement
Compute ##A^j~\text{for} ~~j=1,2,...,n## for the block diagonal matrix##A=\begin{bmatrix}
J_2(1)& \\
&J_3(0)
\end{bmatrix}##,
And show that the difference equation ##x_{j+1}=Ax_{j}## has a solution satisfying ##|x_{j}|\rightarrow\infty~\text{as}~j\rightarrow\infty##...
Homework Statement
[/B]
Change the independent variable from x to θ by x=cosθ and show that the Legendre equation
(1-x^2)(d^2y/dx^2)-2x(dy/dx)+2y=0
becomes
(d^2/dθ^2)+cotθ(dy/dθ)+2y=0
2. Homework Equations
The Attempt at a Solution
[/B]
I did get the exact form of what the equation...
Homework Statement
[/B]Homework Equations
The Attempt at a Solution
I've highlighted two equations on the screenshot. How did it proceed from the first to the second? I'm actually confused with the absolute values. What is the idea behind getting rid of the first absolute value(1-5v^2) while...
Homework Statement
Given the following difference equation;
x(k+2)-x(k+1)+0.25x(k)=u(k+2)
where
x(0)=1; x(1)=2; u(k)= 1 for k=1,2,3,…
Homework Equations
Z- transformation
The Attempt at a Solution
To be able to solve this difference equation, I think I need to transform it into z domain...
This is regarding difference equations and their solutions. And to be specific, this is about linear constant coefficient difference equations. I read at one place, that the general solution of it can be expressed as sum of homogeneous equation and the particular solution. And at another place...
Hi, Please I need some help, how can I get the Laplace transform of the integration of a difference equation??
$\int _{ 0 }^{ \infty }{ { e }^{ -st } } \int _{ -\tau }^{ 0 }{ G(\theta )x(t+\theta )d\theta } dt$
Many thanks in advanced.
hello. I have a MATLAB skeleton provided because i want to model a distribution with a circular geometry. all in all, i want the 3d graph of the code to be some type of cylinder. This is the code:
% flat step condition
for ii=1:nHi,
for jj=1:nHj,
if (X(ii)/R_P)<1 &...
I have the differential equation
\frac{dM}{dt}=4\pi \rho(r,t)r(t)^2\frac{dr}{dt}
which is the first term from
M(t)=4\pi\int_0^{r(t)}C(r,t)r(t)^2dr
This describes the change in mass (M) of a sphere from a change in radius (r) given a density (rho) that depends on radius and time (t).
My...
Homework Statement
The problem is tough to type out correctly. Pasting problem statement image
http://postimg.org/image/a0r92a0wl/
http://postimg.org/image/a0r92a0wl/
The Attempt at a Solution
I just need to know how to proceed with the problem. Not the answer. This is the scan...
Solve the difference equation yn+1=sqrt((n+3)/(n+1)) yn in terms of the initial value y0.
y1=sqrt(3)y0
y2=sqrt(6)y0
yn=sqrt(3n)y0
But that's not the answer. The answer in the textbook is yn=y0sqrt(((n+2)(n+1))/2). Did I do something wrong?
Solve the difference equation yn+1=-0.9yn in terms of the initial value y0.
y1=-0.9y0
y2=-0.9y1=(-0.9)2y0
yn=(-0.9)ny0
Is this the answer? Because the answer in the textbook says yn=(-1)n(0.9)ny0. Please help.
Hello,
I hope someone can help me with a problem I am having. It is neither homework or coursework, but for my own understanding.
I should say from the start, I am one of those people who tend not to be able to see the forset because all the trees are in the way, so I probably will be...
For the formula for getting voltage difference V_b-V_a=-\int _a^{b} Edl how do we know where the limit a and b go? In the equation it goes from a to b but why not b to a? For example , in this question I am given a non uniform charge density where charge density increases with radius r for a...
Homework Statement
Consider your bank offers a 2% annual interest rate, and has no annual service charge. Let y[n] represent your account balance at the beginning of the month n and x[n] represent the amount of money you deposit during the month n. Assume that deposits during month n are...
Homework Statement
Find the digital filter equivalence of the
circuit in a difference equation formHomework Equations
Is this the difference equation form?The Attempt at a Solution
If that is the proper form.. I think this is how to solve it? I got this example from my textbook I'm just not...
Homework Statement
Work out the correct coefficient arrays for these equations:
y(n)=y(n-1)+\frac{1}{5}(x(n)-x(n-5))
y(n) = 0.82y(n -1) + .28x(n)
Homework Equations
\sum a(r)y(n+1-r)=\sum b(k)y(n+1-k) where a(1) = 1
The Attempt at a Solution
Ok for the second equation...
Trying to make a three line loop that would differentiate x2√4x+1 to the nth term starting from the original function (n=0) to the 5th derivative (n=5) and then substitute 2 into the derivatives. Here's what I got
f = @(x) x2√4x+1
For n=[0:5] - As the nth term goes from 0 to 5...