0
votes
0answers
20 views

Solving a simple Recurrence in summation form(very special case)

I have a bit confusing recursion form $\sum_{n=2}^{\infty}\{f(n)\frac{n}{n-1}\}=C, \tag 1$ $f(0)=b,f(1)= a,f(2)=c$ and $C$ are constants. Could you help me to solve this recursion or help me to ...
-1
votes
1answer
52 views

Initial values are lost (diff eq to Transfer function)?

I read eternal Julius O. Smith III and he says that $$x_{n-m} = z^{-m}X(z)$$ Particularly, difference relation $$y_{n} = y_{n-1} + x_{n}$$ is solved by by $$Y = z^{-1}Y + X = {X \over ...
1
vote
0answers
39 views

General Solution to Linear Difference Equation: Is this correct

Notation: Let $$Q_1[f(x)] = \lim _{h \rightarrow 1} \frac{f(x + h) - f(x) }{h}$$ And let $$Q_1^{-1}\left[f(x)\right] = G(x)|Q_1\left[g(x)\right] = f(x)$$ Consider the equation $$a_0(x) + ...
2
votes
1answer
24 views

Uniqueness of solutions to linear recurrence relations

I understand that if I have a linear homogeneous recurrence relation of the form $q_n = c_1 q_{n-1} + c_2 q_{n-2} + \cdots + c_d q_{n-d}$, I can construct the characteristic polynomial $p(t) = t^d - ...
1
vote
3answers
37 views

Finding an exact solution to a difference equation

How would I solve an equation of the form: $u(n+1)=1/2u(n)+(1/3)^n$ when $u(0)=1$? I got an answer of the form $u(n)= c + \sum(1/3)^j*2^{j-1}$ but believe this is incorrect?
0
votes
0answers
25 views

Generic Exponential curve base derivation

Alrighty so I am working on a computer program that forms ADSR envelopes including exponential curves for the attack, decay, and release segments. It uses the following equation for the exponential ...
0
votes
1answer
42 views

2nd Order Homogeneous ODE recurrence relation??

Doing some exam revision and have been stumped by this; the question asks you to find the recurrence relation satisfied by the coefficients. Attempt at solution: I have already found that there ...
2
votes
2answers
54 views

Uniform convergence of matrix integral sequence

I was given recursively defined: $M_k=I+\int_{t_0}^tA(s)M_{k-1}(s)~ds$ and $M_0=I$ and that $A(t)$ is a matrix with entries that are continuous functions on $t_0\leq t\leq t_1$. By induction we can ...
0
votes
1answer
57 views

Can every recurrence relation be solved?

Motivation A possible way to solve an ODE is to express the solution as: $y= \sum_{n=0}^\infty a_nx^n$. We substitute in the ODE and then calculate the coefficients $a_n$. For example, $y''+y=0$ ...
0
votes
3answers
95 views

Solve the recurrence $ T_{n + 1} = T_{n} + nT_{n - 1}$

Solve the recurrence $$ T_{n + 1} = T_{n} + nT_{n - 1}\,, \quad\mbox{for}\quad n \geq 1\quad \mbox{with initial conditions}\ T_{0} = T_{1} = 1 $$ by finding the exponential generating function and ...
1
vote
1answer
732 views

Legendre polynomials recurrence relation

How can i get? $$P_{n+1}=xP_n(x)-\frac{1-x^2}{n+1} P'_n(x)$$ $n>=0$ Also know as the leadder equation of the legendre polinomials i tried to use de recurrence relations as: ...
0
votes
1answer
119 views

Differential Equations: Find the first four terms in each of two solutions y1 and y2 …

The differential equation is $y'' - xy' - y = 0$ with $x_0 = 1$ Now, I know how to find the recurrence relation... and it's given by: $a_(n+2) = [(a_(n+1) + a_(n)) / (n+2)]$ But I can't quite ...
3
votes
3answers
78 views

General solution to a Growth equation

I'd like to compute a formula that describes a population growth. The population starts with $N(t=0)$ individuals. At each time step there are births and deaths. The number of births at time $t$ is ...
5
votes
1answer
72 views

Recurrence-differential equation

In his book on differential equations, Arnold writes that $x'(t)=x(x(t))$ is not a differential equation. My question is: how can one solve it?
0
votes
4answers
560 views

Second-Order, Linear Inhomogeneous Recurrence Relation With Constant Coefficients

How does one solve the general recurrence relation $$s_n=\alpha s_{n-1}+\beta s_{n-2}+\zeta(n)?$$
4
votes
1answer
312 views

The characteristic polynomial of a recurrence relation.

If I have a linear homogeneous recurrence relation $$y_n=c_1y_{n-1}+\ldots+c_ky_{n-k},$$ I can get its characteristic equation, which is $$r^k=c_1r^{k-1}+\ldots+c_k.$$ In particular for ...
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
2answers
54 views

How to solve this inhomogeneous recurrence difference equation?

$a_n=1+p a_{n-1+k} + (1-p) a_{n-1}$, $a_0=0$ Given that $0<p<1$, $n,k$ are positive integers, and $a_n<\infty$ If I am only interested in real value solutions, how to solve it? If there is ...
3
votes
2answers
145 views

How to solve the differential equation $u_k(z)=-2\cfrac{\partial}{\partial z }(\cfrac{u_{k+1}(z)}{z})$?

$$e^{z\sqrt{1-t}}=\sum \limits_{k=0}^\infty \frac{u_k(z)t^k}{k!}$$ $$\frac{\partial}{\partial t }(e^{z\sqrt{1-t}})=\frac{\partial}{\partial t }(\sum \limits_{k=0}^\infty \frac{u_k(z)t^k}{k!})$$ ...
2
votes
1answer
134 views

How can I express such function as known functions or power series?

$$\int_0^x \cfrac{1}{1+\int_0^t \cfrac{1}{2+\int_0^{t_1} \cfrac{1}{3+\int_0^{t_2} \cfrac{1}{\cdots} dt_3} dt_2} dt_1} dt =f(x)$$ $$\int_{0}^{x} \frac{1}{n+h_{n+1}(t)}{d} t=h_n(x)$$ ...
0
votes
0answers
58 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 ...
6
votes
3answers
381 views

Common term for differential equations and recurrence relations

Recently I have been working with recurrence relations (mostly linear), and systems of coupled recurrence relations. I have noticed a lot of common ground with differential equations. In a way, you ...
2
votes
1answer
452 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 ...
3
votes
3answers
236 views

Common theory for Linear equations, linear ODEs and linear recurrence relations

In Linear equations, linear ODEs and linear recurrence relations, when solving homogeneous equations, there are a subspace of solutions, and when solving inhomogeneous equations, a particular ...