(4) Consider the equationdr(t)dt=(205+0.1)et,with initial condition z(0) = 1.0. Find an approximate solution to (1) by evolving the systemfrom t = 0 tot = 1 with a single 2nd-order Runge-Kutta step. Give a rough quantitative estimateof the discretisation error by comparing two different methods or two different step sizes.=

Given differential equation is dxtdt = x0.5 +0.1e-t. Initial condition:- x0=1.0 Find an approximate…

Answered: (4) Consider the equation dz(t)…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Consider the autonomous system
A mass weighing 32 lb is attached to a spring hanging from the ceiling and comes to rest at its equilibrium position. At time t= 0, an external force of F(t) = 3 cos 2t Ib is applied to the system. If the spring constant is 15 Ib/ft and the damping constant is 2 Ib-sec/ft, find the steady-state solution for the system. Use g = 32 ft / sec". The steady-state solution is y(t) =|.
Consider finite difference method to approximation the solution to BVP on 1. 2 y" = -e y(1) = 0 y(2) = In 2 %3D with n 3. Set up the system of equations for unknown wi, and find its Jacobian.
Consider the system Sai = 2x2 2 = -2x1 %3D Subject to the initial conditions x1(0) = 3, x2(0) = 4. (a) Transform the given system into a single equation of second order. (b) Find r1 and xz that also satisfy the given initial conditions (c) Sketch the graph of the solutions in the r182-plane for t > 0.
5
Consider the initial value problem (IVP): 4a + 3a?y2 +2x'y- dy = 0, y(1) = -1. da (a) Solve the initial value problem. (b) Give the largest interval over which the solution of the (IVP) is defined.
Use the Newton Raphson Method with X®)=[-1 -2 1]T to approximate the solution to the given nonlinear system with an error tolerance of &=0.01 in the maximum magnitude norm (|X). x,' + xx, - x,x, = -6 e +e* -x, = 0 x - 2x,x, = 4
Consider a system of differential equations X'= Ax, where X = and the 2x2 matrix A has eigenpairs 1 Vị = 1 d1 = 3, and d2 = 2, V2 V2 From the options below, find the correct behavior of the solution x(t) = -Vị e^it + v2 est. lim ||x(t)|| lim ||x(t)|| = 0, = ∞, t→-∞ x(0) = [va): x(t) x(t) lim t→0 x(t)|| V1, lim V2, t-o ||x(t)|| lim ||x(t)|| = 0, lim ||x(t)|| =∞, t→∞ t -∞ x(t) x(t) lim x(0) V2, lim V1, 00 ||x(t)|| t-0 x(t)|| lim ||x(t)|| = ∞, lim ||x(t)|| = 0, t→∞ x(0) = ,v. x(t) x(t) lim t→-0 ||x(t)|| lim V1, V2, t>0 ||x(t)|| lim ||x(t)|| = 0, lim ||x(t)|| = 0∞, t→∞

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