Consider the following ODE dN + N = u(t-1) - uſt - 3) with initial condition...
Consider the following initial value problem. y′ + 5y = { 0 t ≤ 1 10 1 ≤ t < 6 0 6 ≤ t < ∞ y(0) = 4 (a) Find the Laplace transform of the right hand side of the above differential equation. (b) Let y(t) denote the solution to the above differential equation, and let Y((s) denote the Laplace transform of y(t). Find Y(s). (c) By taking the inverse Laplace transform of your answer to (b), the...
a can be skipped Consider the following second-order ODE representing a spring-mass-damper system for zero initial conditions (forced response): 2x + 2x + x=u, x(0) = 0, *(0) = 0 where u is the Unit Step Function (of magnitude 1). a. Use MATLAB to obtain an analytical solution x(t) for the differential equation, using the Laplace Transforms approach (do not use DSOLVE). Obtain the analytical expression for x(t). Also obtain a plot of .x(t) (for a simulation of 14 seconds)...
(1 point) Consider the initial value problem d'y dy dt2 dt dt Write down the Laplace transform of the left-hand side of the equation given the initial conditions Your answer should be a function of s and Y with Y denoting the Laplace transform of the solution y Write down the Laplace transform of the right-hand side of the equation Your answer should be a function of s only. Next equate your last two answers and solve for Y. You...
2. Consider the following 1-D wave equation with initial condition u (x, 0)- F (x) where F(x) is a given function. a) Show that u (x, t)-F (x - t) is a solution to the given PDE. b) If the function F is given as 1; x< 10 x > 10 u(x, 0) = F(x) = use part (a) to write the solution u(x, t) c) Sketch u(x,0) and u(x,1) on the same u-versus-x graph d) Explain in your own...
Consider the following second-order ODE representing a spring-mass-damper system for zero initial conditions (forced response): where u is the Unit Step Function (of magnitude 1 a. Use MATLAB to obtain an analytical solution x() for the differential equation, using the Laplace Transforms approach (do not use DSOLVE). Obtain the analytical expression for ao. Also obtain a plot of x() (for a simulation of 14 seconds) b. Obtain the Transfer Function representation for the system. c. Use MATLAB to obtain the...
pls do all questions. thanx 1. [5 Consider the IVP rty(t) + 2 sin(t)y(t) = tan(t) y(5)=2 Does a unique solution of the IVP exist? Do not solve the IVP but fully justify you answer. What is the IOE? 2. 4 Consider the ODE Using undetermined coefficients, what is an approprite guess for the coefficient (s) in yp but fully justify you answer. ? Do not solve for 3. [10] Solve the IVP. Use any approach you like y(x) 6y'(x)...
3. Consider the Linear Time-Invariant (LTI) system decribed by the following differential equation: dy +504 + 4y = u(t) dt dt where y(t) is the output of the system and u(t) is the input. This is an Initial Value Problem (IVP) with initial conditions y(0) = 0, y = 0. Also by setting u(t) = (t) an input 8(t) is given to the system, where 8(t) is the unit impulse function. a. Write a function F(s) for a function f(t)...
please provide me with full working solution. Any help is appreciated. thank you in advance Consider the diffusion equation, au(x,t u(x,t) Here u(x,t) > 0 is the concentration of some diffusing substance, the spatial variable is x, time is t and D is a constant called the diffusivity with dimensions [LT-11. We will consider the diffusion equation on a finite spatial domain (0<x< 1) and an infinite time horizon (t > 0). To solve the diffusion equation we must include...
Please answer all questions and Please Write Clearly Problem #1 : (14%) Consider the following differential equation: 0.2' dc(t) + x(t)-21,(t) with initial condition x(0)=1 and u (1) being a unit step function. dt la) Convert the differential equation into a Laplace transformed algebraic equation in which X'(s) is the Laplace transform of x(t). (3%) (lb) Solve the algebraic equation for X(s). (3%) (1 c) Find the inverse laplace transform of x(s) , which is the solution of the differential...
Use the Laplace Transform to find the solution of the initial value problem fort > 0. No credit will be awarded for other methods. You way write your solution in terms of the unit step function U(t) and its translates. y + 4y = 2t . Uſt - 1), y(0) = 0.