Use Laplace transformations to solve the following ODE for (t): ä(t) + 2r(t) = u(t) +...
6. Solve an ODE Using Laplace Transforms: For this problem you are to use Laplace Transforms. Find the complete solution for the initial value problem yº+w2y = t +u.(t - Ttcost, y(0) = 1, y(0) = 0. Hint: Look carefully at the second forcing term and rewrite cost. You can solve this by brute force using the integral below. It would be a good exercise to make sure both approaches give the same Laplace transform. The integral The solution ſeat...
Consider the following ODE dN + N = u(t-1) - uſt - 3) with initial condition N(0) = 0, where u is the unit step function. Sketch the function on the right-hand side of the ODE. Now use a Laplace Transform to solve the IVP. Sketch your solution for N(t).
Solve Utilizing Laplace Transformations: 3y" + 3y' + 6y = 3e^(-t) * sin2t with initial conditions y(0) = 1 and y'(0) = -1
Second order systems of ordinary differential equations (ODE) often describe motional systems involving multiple masses. Solve the following second order system of ODE using Laplace transform method: Xy-=5x1-2x2 + Mu(t-1) x2-=-2x1 + 2x2 x,(t) and x2(t) refer to the motions of the two masses. Consider these initial conditions: x1 (0) = 1, x; (0)-0, x2(0) = 3, x(0) 0 Second order systems of ordinary differential equations (ODE) often describe motional systems involving multiple masses. Solve the following second order system...
Solve the following ode using Laplace transform: y' - 5y = f(t); y(0) - 1 t; Ost<1 f(t) = 0; t21
A system is modeled by the following LTI ODE: ä(t) +5.1640.j(t) + 106.6667x(t) = u(t) where u(t) is the input, and the outputs yı(t) and yz(t) are given by yı(t) = x(t) – 2:i(t), yz(t) = 5ä(t) 1. Find the system's characteristic equation 2. Find the system's damping ratio, natural frequency, and settling time 3. Find the system's homogeneous solution, x(t), if x(0) = 0 and i(0) = 1 4. Find ALL system transfer function(s) 5. Find the pole(s) (if...
please make sure u solve in clear steps and 100% correct 4. (21 pts) Laplace Transforms of ODE-A mechanical system has the following 2nd order differential equation describing its position x(t): d+x(e) – 4 dx(t) + 4x(t) = 0. The initial conditions are: x'= 3.9m/s and x(@) = 2.1m a. (3 pts) Convert the differential equation into the s-domain. Substitute in the initial conditions as needed XCS/ dt2dt
Determine the Laplace transform of x(t) = t2 u(t – 1) (b) Use Laplace transform to solve the following differential equation for t ≥ 0. ? 2?(?) ?? 2 + 3 ??(?) ?? + 2?(?) = (? −? ????)?(?); ?(0) = 1; ??(0) ?? = −3
Problem 2: Consider the following differential equation: 0 and with u = e-31. Solve for x(t) using with initial conditions x(0)-x(0) Laplace transforms.
Find the following Inverse Laplace transformations. Use the Laplace Transform table attached in the next page. Show all your work, how to get partial fractions etc. and clearly state the Laplace rule(s) that you used in the related step from the attached Laplace Table. (?) ℒ −1 { ? 2−?+2 ?(?−3)(?+2) } (?) ℒ −1 { ? −? ? ? } (?) ℒ −1 { 1 ? 2−2?+1 }. Q1. (15 pts) Find the following Laplace transformations. Use the Laplace...