# Tagged Questions

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### 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 ...
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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 $p(t) = t^d - ... 3answers 37 views ### Finding an exact solution to a difference equation How would I solve an equation of the form:$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}$but believe this is incorrect? 0answers 27 views ### Generic Exponential curve base derivation Alrighty so I am working on a computer program that forms ADSR envelopes including exponential curves for the attack, decay, and release segments. It uses the following equation for the exponential ... 1answer 42 views ### 2nd Order Homogeneous ODE recurrence relation?? Doing some exam revision and have been stumped by this; the question asks you to find the recurrence relation satisfied by the coefficients. Attempt at solution: I have already found that there ... 3answers 69 views ### Uniform convergence of matrix integral sequence I was given recursively defined:$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$. By induction we can ... 1answer 60 views ### Can every recurrence relation be solved? Motivation A possible way to solve an ODE is to express the solution as:$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$... 3answers 96 views ### Solve the recurrence$ 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 ... 1answer 859 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$Also know as the leadder equation of the legendre polinomials i tried to use de recurrence relations as: ... 1answer 119 views ### Differential Equations: Find the first four terms in each of two solutions y1 and y2 … The differential equation is$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)]$But I can't quite ... 3answers 81 views ### General solution to a Growth equation I'd like to compute a formula that describes a population growth. The population starts with$N(t=0)$individuals. At each time step there are births and deaths. The number of births at time$t$is ... 1answer 73 views ### Recurrence-differential equation In his book on differential equations, Arnold writes that$x'(t)=x(x(t))$is not a differential equation. My question is: how can one solve it? 4answers 591 views ### Second-Order, Linear Inhomogeneous Recurrence Relation With Constant Coefficients How does one solve the general recurrence relation $$s_n=\alpha s_{n-1}+\beta s_{n-2}+\zeta(n)?$$ 1answer 337 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 ... 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 ... 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 ... 2answers 145 views ### How to solve the differential equation$u_k(z)=-2\cfrac{\partial}{\partial z }(\cfrac{u_{k+1}(z)}{z})\$?

$$e^{z\sqrt{1-t}}=\sum \limits_{k=0}^\infty \frac{u_k(z)t^k}{k!}$$ $$\frac{\partial}{\partial t }(e^{z\sqrt{1-t}})=\frac{\partial}{\partial t }(\sum \limits_{k=0}^\infty \frac{u_k(z)t^k}{k!})$$ ...
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### How can I express such function as known functions or power series?

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