5, Compute (f *g)(t) where f(t) = t and g(t) = et by using the definition...
8. (i) Find C[F(t)], where F(t) = { if 0 st 34, ift> 4 (ii) Compute the convolution e2 et directly by the definition of the convolution (iii) Evaluate Lle-2445 - e cos(4t) + sin(V2t)). blom.
4. Use the convolution integral to find f, where f = g*h, and g(t) = et ult) h(t) = e-2t u(t) Note that both of these are causal to simplify the integration.
Consider f(t) = cos(9t), g(t) = et. Proceed as in this example and find the convolution f ∗ g of the given functions.
Problem 4 Given: St t(t) # -t e g(t) a) Compute fg () using convolution integral method. b) Compute g*f () with Laplace transform. o) What are the differences between the results of questions (a) and (0) above? d) Find the Laplace transform of the following function: (t 0 to +oo) e dt e) Find the equivalent solution of (d) using MATLAB method) (find 2 methods)
Problem 4 Given: St t(t) # -t e g(t) a) Compute fg () using...
) Verify your formula for 1"n directly from the definition of the convolution product by using induction on n and a calculation like: *T et
) Verify your formula for 1"n directly from the definition of the convolution product by using induction on n and a calculation like: *T et
(3 + 2ry,12-3уг), 5. Find JeF . Tds, (et sin t, et cost),0 t where F(x,y) and C is given by r(t) - T.
(3 + 2ry,12-3уг), 5. Find JeF . Tds, (et sin t, et cost),0 t where F(x,y) and C is given by r(t) - T.
Question 1: (25 marks) Compute the convolution f(t) = fi(t) + f(t) of the following two time-functions fi(t) and fu(t). Provide a complete analytical solution (formula) and also a plot of your final result f(t). You are allowed to use, if you wish, any of the properties of convolution that we have studied, but you need to explain clearly how/where you have used them. fit) f2t) +3 +4 +5 -1 0 + 1 t -1 0 +1 +2
6. (20 pts) se the definition of convolution to compute tu(t) * u(t); Check your answer using Laplace transform. 7. (40 pts) Solve the following ODE. y" 4y 4y = e-4*[u(t) - u(t - 1)] (0) = 0; y'(0)= -1
Find the convolution f(t) *g(t) for the following problem. f(t) = g(t) = 9 sint (f*g)(t) =
[12 marks] Using the definition of big-O, show that f(x) is big-O of g, where: f(x) = 2* + 33 and g(x) = 3* Show the details of your work to obtain a full mark.