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(15 points) This problem is related to Problem 7.23-24 in the text. Given the differential equati...
Given the system of differential equations o y (7tcos(tut) Write the first order matrix differential equation that is the basis for using Euler's method to compute the numerical solution. It is assumed you will use two auxiliary functions, xi and t2 Define the functions i and 2 in terms of v and y. E2 dri (t) dt 1(t) dr2(t) dt a2(t) Given the system of differential equations o y (7tcos(tut) Write the first order matrix differential equation that is the...
(15 points) This problem is related to Problems 7.5-7.8 in the text. Given the differential equation y' + 5y = 9cos(5t + 0.523599)u(t). Remember to use u(t) in Webwork answers when the solution is for t > 0. For differential equations, angles are given in radians. a. Find the complementary solution, yc(t), with constants denoted C1, C2, ... yc(t) = help (formulas) b. Find the particular solution, yp(t). yp(t) = help (formulas) c. Find the total solution, y(t) for the...
please answer b. and c. Problem 1. Consider the differential equation given by (a) On the axes provided below, sketch a slope field for the given differential equation at the nine points indicated. locales de mor t e wold qolution to the given differential equation with the initial condition (b) Let y = f(x) be the particular solution to the given differential equation with the initial condition f(0) = 3. Use Euler's method starting at x = 0, with a...
12. Find all the equilibrium points of the competing species model [2.5 marks] x' = 0.6-0.2x2-0. Iy 13. Select the second-order equation "+9xy+6y 2x2 written as a system of first-order 2.5 marks] equations a) u'=v' b) uv u'=v v'=-9xv-би-2x2 d) v d) 4xy 14. Consider the differential equation given by y a) On the axis provided below, sketch a slope field for the given differential equation at the five points indicated by a circle. 2 marks] b) Use Euler's method...
Consider the following differential equation which describes a spring-mass-damper system më + ci + kx = cos(2nt) where c 1.9 and k = 3.1. The initial conditions are given as x(0) = 0 and 2(0) = 0 and the time step is 0.1 s. 1. Assuming that m - 0, use the Runge-Kutta 4th order method to find (a) x(0.1) and (b) *(0.1). 2. Assuming that m 1, use Euler's method to find (a) 2(0.2) and (b) X(0.3).
(10 points) This problem is related to Problems 8.16-21 in the text. Consider the differential equation y" (t) + 16y'(t) + 68y(t) = –20e-4t u(t), with initial conditions y(0) = -3, and y'(0) = 4. Find the Laplace transform of the solution Y(s). Write the solution as a single fraction in s Y(s) = help (formulas) Find the partial fraction decomposition of Y(s). Enter all factors as first order terms in s, that is, all terms should be of the...
Find the value of x(0.5) for the initial value problem at = thx(0)=1 using Euler's method with step size h 0.05 Find the value of x(0.4) for the coupled first order differential equations together with initial conditions with step size 0.1: 2. dt t+x 3. dx dt = y, dy dt x(0) = 1.2 and --ty +xt2 + y(o) 0.8 Find the value of x(0.5) for the initial value problem at = thx(0)=1 using Euler's method with step size h...
Problem Thre: 125 points) Consider the following initial value problem: dy-2y+ t The y(0) -1 ea dt ical solution of the differential equation is: y(O)(2-2t+3e-2+1)y fr exoc the differential equation numerically over the interval 0 s i s 2.0 and a step size h At 0.5.A Apply the following Runge-Kutta methods for each of the step. (show your calculations) i. [0.0 0.5: Euler method ii. [0.5 1.0]: Heun method. ii. [1.0 1.5): Midpoint method. iv. [1.5 2.0): 4h RK method...
Problem 3. Given the initial conditions, y(0) from t- 0 to 4: and y (0 0, solve the following initial-value problem d2 dt Obtain your solution with (a) Euler's method and (b) the fourth-order RK method. In both cases, use a step size of 0.1. Plot both solutions on the same graph along with the exact solution y- cos(3t). Note: show the hand calculations for t-0.1 and 0.2, for remaining work use the MATLAB files provided in the lectures Problem...
Problem #18: [2 marks] Let W be the subspace of R4 spanned by the vectors u - (1,0,1,0), u2 = (0.-1, 1.0), and ug = (0.0, 1,-1). Use the Gram-Schmidt process to transform the basis (uj, u, uz) into an orthonormal basi (A) v1 = (-12,0, 2.0), v2 - (VG VG VG, o), v3 - (I ) (B) v1 = (-V2.0, .), v2 - (VG VG VG o), v3 - (™J - V3 VI-V3) (C) v1 - ($2.0, 92.0), v2...