g is [0,infinity) Question 6 (15 points) Let be a piecewise continuous function of exponential order...
Let g be a piecewise continuous function of exponential order on [0, 0). Use the Laplace transform to solve the following initial value problem. dy -(t) + 2y(t) = g(t), y(0) = 0, dy (0) = 1. Express your answer by using the convolution operator *.
For full credit, you must show all work and box answers 1. If functions f and g are piecewise continuous on the interval [0, oo), then the convolution of f and g is a function defined by the integral The Convolution Theorem (theorem 7.4.2 in your book and formula 6 in your table) states: If j(t) and g) are piecewise continuous on [0, oo) and of exponential order, then We are going to use convolution to solve y"-y,-t-e-,, y(0)-0, y'(0)-0....
6) True or False? (justify your answers a) I f ft) is piece wise Continuous on [goo) and of exponential order and L [f(t)] = FC), then L [ S t f (G) I TE F(S) ? S 6) The Function F(s) = 1 is the Laplace transform of a function that is a piecewise continuous on [o,oo) and of exponential order?
Question 9 3 pts The Laplace transform of the piecewise continuous function J4, 0< < 3 f(t) is given by 2, t> 3 2 L{f} (2 - e-st), 8 >0. S L{f} (1 – 3e-), 8>0. 8 2 L{f} (3 - e-s), 8 >0. S L{f} = (1 – 2e-st), s > 0. None of them Question 10 3 pts yll - 4y = 16 cos 2t To find the solution of the Initial-Value Problem y(0) = 0 the y...
Question 9 3 pts The Laplace transform of the piecewise continuous function 4, 0<t <3 f(t) is given by t> 3 (2, L{f} = { (1 – 3e-*), s>0. O 2 L{f} (2 - e-st), 8 >0. 2 L{f} = (3 - e-st), s >0. O None of them 1 L{f} (1 – 2e -st), s >0.
(1 point) Consider the initial value problem d2y dy 8 +41y8 cos(2t), dt dy (0) y(0) = -2 -6 dt dt2 Write down the Laplace transform of the left-hand side of the equation given the initial conditions (sA2-8s+41)Y+2s-18 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 (-8s+32)/(sA2-8s+20) Your answer should be a function of s only...
Consider the function: 2) g(t) = tet"sin (et a)ls g(t) continuos at [0,°°] and of exponential order? b)Show that Laplace transform exist fot Re(s)>0
Consider the function: 2) g(t) = tet"sin (et a)ls g(t) continuos at [0,°°] and of exponential order? b)Show that Laplace transform exist fot Re(s)>0
Please help both questions, thanks
(1 point) Let g(t) = e2 a Solve the initial value problem 4 – 2 = g(t), using the technique of integrating factors. (Do not use Laplace transforms.) y(0) = 0, (t) = b. Use Laplace transforms to determine the transfer function (t) given the initial value problem 6' - 24 = 8(t), (0) = 0. $(t) = c. Evaluate the convolution integral (6 + 9)(t) = Sølt – w)g(w) dw, and compare the resulting...
a) i. Express in terms of the unit step function, the piecewise continuous causal functions (2t2, Ost<3 F(t) = {t + 4, 3 st<5 9, t25 [3 marks] ii. Use Laplace transforms to solve the initial value problem a) 7" + 16y = 4cos3t + s(t – 1/3) where y(0) = 0 and y'(0) = 0. [7 Marks) E.K. Donkoh (Ph.D) or [7 marks) B) y' – 3y = F(t), where y(0) = 0 and (sint, Osts F(t) = 1,...
Answer C
6. Let f be a continuous function on [0, oo) such that 0 f(z) Cl- for some C,e> 0, and let a = fo° f(x) da. (The estimate on f implies the convergence of this integral.) Let fk(x) = kf(ka) a. Show that lim00 fk(x) = 0 for all r > 0 and that the convergence is uniform on [8, oo) for any 6> 0. b. Show that limk00 So ()dz = a. c. Show that lim00 So...