1. Consider the difference equation (1) Sn+2 - -8n+1Sn with initial conditions so = 0, 81...
Expand the function is in a power series anx" with center c = 0. Find anx”. n=0 (Express numbers in exact form. Use symbolic notation and fractions where needed. For alternating series, include a factor of the form (-1)" in your answer.) (-6) anx" = 5n+2 Determine the interval of convergence. (Give your answers as intervals in the form (*, *). Use symbol oo for infinity, U for combining intervals, and appropriate type of parenthesis " (", ")", "[" or...
Problem 3. Consider the following difference equation. 2y[n] + 4y[n – 1] + 10y[n – 2] = u[n] - u[n – 1] – Zu[n – 2] (a) Find the transfer function, H(z). (b) Find the general form of impulse response, h(t). (Do not need to evaluate the actual constants. Leave the constants as ki, k2, etc. The answer should be in its final form without the complex numbers.) (c) Find the general form of step response, y(t). (Do not need...
Consider the following differential equation. (1 + 5x2) y′′ − 8xy′ − 6y = 0 (a) If you were to look for a power series solution about x0 = 0, i.e., of the form ∞ Σ n=0 cn xn then the recurrence formula for the coefficients would be given by ck+2 = g(k) ck , k ≥ 2. Enter the function g(k) into the answer box below. (b) Find the solution to the above differential equation with initial conditions y(0) ...
Consider the below wave equation with the given conditions. au 81 Ox? u(0,1) het au 0 < x < 4, t > 0, u(4,t) = 0, 1 > 0 op u(x,0) = 0, ди at = 6x(4- x) = 384 ${1 - (-1)"} sin(npox/4), 0< x < 4. n=1 The solution to the above boundary-value problem is of the form u(x,t) = 8(n, t) sin "* n=1 Find the function g(n,1).
In this exercise we consider finding the first five coefficients in the series solution of the first order linear initial value problem (+3)y' 2y 0 subject to the initial condition y(0) 1. Since the equation has an ordinary point at z 0 it has a power series solution in the form We learned how to easily solve problems like this separation of variables but here we want to consider the power series method (1) Insert the formal power series into...
4. Define the function f(x) = x/(e* – 1) so that f(0) = 1, then the Maclaurin expansion of f(x) has a positive radius of convergence. a) When the Maclaurin expansion is expressed in the form B2 B3 3 i=1+ B13 +31 x2 + x3 + ... + x" + ... n! the coefficients B1, B2, B3,... are called the Bernoulli numbers. Substitute the Maclaurin series for et and use polynomial long-division to find the first four Bernoulli numbers. b)...
Consider the following initial value problem, (1 - 2)" + 3xy' - 8y = 0, 3(0) = 3, 7(0) = 0. Note: For each part below you must give your answers in terms of fractions (as appropriate), not decimals. (a) This differential equation has singular points at Note: You must use a semicolon here to separate your answers (b) Since there is no singular point at x = 0, you can find a normal power series solution for y() about...
Problem 1: Consider the following complex numbers: Z1 = 2 + j4 Z2 = 5e3 a) Write zi in polar form b) Write ze in rectangular form c) Plot the product of zı & z2 in the complex plane with x(0) = 0.3 Problem 2: Consider the system: dx/dt = -3x a) Solve the initial value problem b) Plot the resulting function
e differential equation y 0 + y = 1 2−x with the initial conditions y(0) = 2. We wish to approximate y(1) using another method. please help me, thanks so much Consider the same differential equation y' +y= with the initial conditions y(0) = 2. We wish to approximate y(1) using another method. (a) Use the method of series to by hand to find the recursion relation that defines y(t) = 2*, QmI" as a solution to this differential equation....
(1 point) Solve the wave equation with fixed endpoints and the given initial displacement and velocity. a2 ,0<x<L, t > 0 a(0. t) = 0, u(L, t) = 0, t > 0 Ou Ot ηπα t) + B,, sin (m Now we can solve the PDE using the series solution u(r,t)-> An C computed many times: An example: t) ) sin (-1 ). The coefficients .An and i, are Fourier coefficients we have , cos n-1 sin(n pix/ L) dr...