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2. Transform of Periodic Functions (a) Determine (t)) where (t) is the square wave function shown...
Use Laplace Transform to solve this problem: Determine the solution of the following initial value problem...
Use Laplace Transform to solve this problem: Determine the solution of the following initial value problem y"(t) + 3y'(t) + 2y(t) = f(t), y(0) = 0, y'(0) = 0. where f(t) = 1 when t lies between 2n and 2n + 1, and f(t) = 0 when t lies between 2n – 1 and 2n (for non-negative integers n). This is a square-wave forcing term.
2. Use the Laplace Transform to solve the initial value problem y"-3y'+2y=h(t), y(O)=0, y'(0)=0, where h...
2. Use the Laplace Transform to solve the initial value problem y"-3y'+2y=h(t), y(O)=0, y'(0)=0, where h (t) = { 0,0<t<4 2, t>4
Consider the initial value problem y" +3y' +2y = (t-1)+r(t), y(0) = y(0) = 0, where...
Consider the initial value problem y" +3y' +2y = (t-1)+r(t), y(0) = y(0) = 0, where 8(t-1) is Dirac's delta function and S4 if 0<t<1 r(t) 8 if t > 1 (a) Represent r(t) using unit step functions. (b) Find the Laplace Transform of 8(t-1)+r(t). (c) Solve the above initial value problem. {
0<t<T when Tt< 2 t 2T sin t when 2. Calculate the Laplace transform of the periodic function f(t) 0 f(t-...
0<t<T when Tt< 2 t 2T sin t when 2. Calculate the Laplace transform of the periodic function f(t) 0 f(t-2) when -7s 3. Calculate the inverse Laplace transform of G(s) 3-4e-5 + $2+2s+17 4. Use the Laplace transform to solve each initial value problem: 4y"+ y u2m(t)sin(t/2) y(0)=0 &(0 =0 (a) 0 and /(0) 2 "+4y+13y = 4to(t-T) if y(0) (b) 5. Use the convolution to write a solution of each initial value problem. y"+6y'+10y g(t) 1 y(0) 0...
10. Use the Laplace transform to solve y" - 3y' +2y f(t), y(0)-0,'(0) 0, where (t)-(0...
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)
Determine the Fourier series expressions for the periodic voltage functions for the full wave rectified sine...
Determine the Fourier series expressions for the periodic voltage functions for the full wave rectified sine wave shown in Figure b and the half wave rectified sine wave shown in Figure c. v(t) 0 2T 3T -T
Af(t) Use the method of Laplace transforms and the accompanying proof results to solve the initial...
Af(t) Use the method of Laplace transforms and the accompanying proof results to solve the initial value problem y" + 3y' + 2y = f(t); y(0)= 0, y'(0)=0 Here, f(t) is the periodic function defined in the graph to the right. Click the icon to review the results of a proof. Square wave Choose the correct answer to the initial value problem below. O A. y(t) = {(1-c-12 OB. y(t) = (1 + e -21) (t + 1)(t+2) 00 O...
a) i. Express in terms of the unit step function, the piecewise continuous causal functions (2t2,...
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,...
Problem (3) a) A periodic square wave signal x(t) is shown below, it is required to answer the below questions: x(t) 1. What is the period and the duration of such a signal? 2. Determine the fund...
Problem (3) a) A periodic square wave signal x(t) is shown below, it is required to answer the below questions: x(t) 1. What is the period and the duration of such a signal? 2. Determine the fundamental frequency. 3. Calculate the Trigonometric Fourier Series and sketch the amplitude spectrum and phase spectrum of the signal x(t) for the first 5 harmonics. b) Find the Continuous Time Fourier Series (CTFS) and Continuous Time Fourier Transform (CTFT) of the following periodic signals...
Q1 Write the following function in terms of unit step functions. Hence, find its Laplace transform...
Q1 Write the following function in terms of unit step functions. Hence, find its Laplace transform 10<tsI g(t) = le-3, +1 , 1<t 2 .22 Q2 Use Laplace transform to solve the following initial value problem: yty(o)-0 and y (0)-2 A function f(x) is periodic of period 2π and is defined by Q3 Sketch the graph of f(x) from x-2t to2 and prove that 2sinh π11 f(x)- Q4 Consider the function f(x)=2x, 0<x<1 Find the a Fourier cosine series b)...
Use Laplace Transform to solve this problem: Determine the solution of the following initial value problem y"(t) + 3y'(t) + 2y(t) = f(t), y(0) = 0, y'(0) = 0. where f(t) = 1 when t lies between 2n and 2n + 1, and f(t) = 0 when t lies between 2n – 1 and 2n (for non-negative integers n). This is a square-wave forcing term.
2. Use the Laplace Transform to solve the initial value problem y"-3y'+2y=h(t), y(O)=0, y'(0)=0, where h (t) = { 0,0<t<4 2, t>4
Consider the initial value problem y" +3y' +2y = (t-1)+r(t), y(0) = y(0) = 0, where 8(t-1) is Dirac's delta function and S4 if 0<t<1 r(t) 8 if t > 1 (a) Represent r(t) using unit step functions. (b) Find the Laplace Transform of 8(t-1)+r(t). (c) Solve the above initial value problem. {
0<t<T when Tt< 2 t 2T sin t when 2. Calculate the Laplace transform of the periodic function f(t) 0 f(t-2) when -7s 3. Calculate the inverse Laplace transform of G(s) 3-4e-5 + $2+2s+17 4. Use the Laplace transform to solve each initial value problem: 4y"+ y u2m(t)sin(t/2) y(0)=0 &(0 =0 (a) 0 and /(0) 2 "+4y+13y = 4to(t-T) if y(0) (b) 5. Use the convolution to write a solution of each initial value problem. y"+6y'+10y g(t) 1 y(0) 0...
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)
Determine the Fourier series expressions for the periodic voltage functions for the full wave rectified sine wave shown in Figure b and the half wave rectified sine wave shown in Figure c. v(t) 0 2T 3T -T
Af(t) Use the method of Laplace transforms and the accompanying proof results to solve the initial value problem y" + 3y' + 2y = f(t); y(0)= 0, y'(0)=0 Here, f(t) is the periodic function defined in the graph to the right. Click the icon to review the results of a proof. Square wave Choose the correct answer to the initial value problem below. O A. y(t) = {(1-c-12 OB. y(t) = (1 + e -21) (t + 1)(t+2) 00 O...
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,...
Problem (3) a) A periodic square wave signal x(t) is shown below, it is required to answer the below questions: x(t) 1. What is the period and the duration of such a signal? 2. Determine the fundamental frequency. 3. Calculate the Trigonometric Fourier Series and sketch the amplitude spectrum and phase spectrum of the signal x(t) for the first 5 harmonics. b) Find the Continuous Time Fourier Series (CTFS) and Continuous Time Fourier Transform (CTFT) of the following periodic signals...
Q1 Write the following function in terms of unit step functions. Hence, find its Laplace transform 10<tsI g(t) = le-3, +1 , 1<t 2 .22 Q2 Use Laplace transform to solve the following initial value problem: yty(o)-0 and y (0)-2 A function f(x) is periodic of period 2π and is defined by Q3 Sketch the graph of f(x) from x-2t to2 and prove that 2sinh π11 f(x)- Q4 Consider the function f(x)=2x, 0<x<1 Find the a Fourier cosine series b)...
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