Solve where .
Step 1:
Find the solution for the homogeneous equation
By integrating, we obtain
, Now to get , we have
Step 2:
Find a particular solution .
Then the general solution is given by .
The solution is diminishing (becomes zero function) as T is decreasing from 1 to 0.1 to 0.01 to 0.001.
+ 2y = 4u, y(0) = 0, for the following input: Solve: dt 0<t<T u(t) t>T...
Problem 2. (a) Solve the initial value problem I y' + 2y = g(t), 1 y(0) = 0, where where | 1 if t < 1, g(t) = { 10 if t > 1 (t) = { for all t. Is this solution unique for all time? Is it unique for any time? Does this contradict the existence and uniqueness theorem? Explain. (b) If the initial condition y(0) = 0 were replaced with y(1) = 0, would there necessarily be...
5. Given y,-t is a solution of t2y" + 2t1-2y = 0, for t > 0, find a second solution y2.
Solve y'' +9y = $(t – 6), y(0) = y'(0) = 0 g(t) = for t < 6 for t > 6
Solve y'' + 4y = $(t – 6), y(0) = y'(0) = 0 g(t) = for t < 6 fort > 6
Solve for y(t). dy/dt + 2x = et dx/dt-2y= 1 +t when x(0) = 1, y(0) = 2
Solve the equation yu- xui = u, t > 0,x >0 with the initial conditions u(x, 0) =1 + x2 using the method of characteristics. Find the u(x, y). Substitute your found solution u(x, y) in the equation and verify that it satisfies the equation. solution explicitly in the form u =
10. Use the Laplace transform to solve y" - 3y' +2y f(t), y(0)-0,'(0) 0, where (t)-(0 for 0 st < 4; for t 2 4 No credit will be given for any other method. (10 marks)
dt - Solve the following equation for y(t) using Fourier Transforms. dy(t) ? +2y(t) = { 'h(t) where h(t) is the Heaviside function: (0,t=0 h(t)= | 1,20 Note: the solution satisfies ly(t) >0 as t →+00.
Consider the signal x(t) = te-atu(t), a > 0 Find to = 1.00 /*(t)?|dt Find to = 10lx(t)2|dt Can simplify to → %*tx?(t)dt x2(t)dt
Page 4 IV. Use the Laplace transform to solve the IVP y' - 2y + y = f(t), y(0) = 1, v/(0) = 1, where (10) 0, t <3 f(t) = t-3, 3 You may use the partial fraction decomposition 16–25+1) 5+(9–1 = (-) + ? + - , but you need to show all the steps needed to arrive to the expression - 022-28+1) in order to receive credit.