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Each of the following equation represents an unforced damped oscillator. Write the Laplace transform of the...
Consider the following initial value problem, representing the response of a damped oscillator subject to the...
Consider the following initial value problem, representing the response of a damped oscillator subject to the discontinuous applied force f(t): y" +2y +10y = f(t), y(0) = 6, 7(0) = -3, f(t) = (1 3<t<4, 10 otherwise. {o In the following parts, use h(t -c) for the Heaviside function he(t) when necessary. a. First, compute the Laplace transform of f(t). L{f(t)}(s) = b. Next, take the Laplace transform of the left-hand-side of the differential equation, set it equal to your...
(write After use Laplace Transform to transform the following initial value problem x" + 3x' +...
(write After use Laplace Transform to transform the following initial value problem x" + 3x' + 2x=2e-t, x(O) = x'(0)=0, you should get X(s)= S-2 fraction as (S-2)/(S-4)(s+6) for (s-4)(3+6) -). Then, find x(t) = L-2(x(s)= 5 (write 5/6 by 6 -3t e^{-3t} by e and sin(2t) or cos(3t) by sin(2t) or cos(3t)).
(write After use Laplace Transform to transform the following initial value problem x" + 2x' +x=3,...
(write After use Laplace Transform to transform the following initial value problem x" + 2x' +x=3, x(0)=0,x'(0)=1, you should get X(s)= S-2 fraction as (S-2)/(S-4)(8+6) for -). Then, find x(t) = £-2(x(s)= (s-4)(3+6) (write 5/6 by 5 -3t 6' , e^{-3t} by e and sin(2t) or cos(3t) by sin(2t) or cos(3t)).
Solve each differential equation. (Don't use the Laplace transform. 3. IVP: y + cos(x + y)...
Solve each differential equation. (Don't use the Laplace transform. 3. IVP: y + cos(x + y) + (x – y + cos(x + y)) = 0, y(0) = 7. If the equation is exact equation, then solve it. If not, find only an exact equation.
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)...
Consider the following nonlinear differential equation, which models the unforced, undamped motio...
Consider the following nonlinear differential equation, which models the unforced, undamped motion of a "soft" spring that does not obey Hooke's Law. (Here x denotes the position of a block attached to the spring, and the primes denote derivatives with respect to time t.) Note: x means x cubed notx a. Transform the second-order de. above into an equivalent system of first-order de.s. b. Use MATLAB's ode45 solver to generate a merical solution of this system aver the interval 0-t-6π...
Find the Laplace Transform for each of the following: 1. L{2sin x + 3e0s 22}= (W)...
Find the Laplace Transform for each of the following: 1. L{2sin x + 3e0s 22}= (W) *** ** (m 3 to the (s? +1)(s2 + 4) 2. 1{eosusa)= ) og 2 ( 6+23+25 ( 6+2+25 2s ZS 3. Find the inverse Laplace Transform L'{- S +1 (4) tsint (B) ’sint (0) (D) rcost
QUESTION 2 use to the following initial value problem (write fraction as (s- After Laplace Transform...
QUESTION 2 use to the following initial value problem (write fraction as (s- After Laplace Transform transform x" + 2x' +x=3, x(O)=0,x'(0)=1, you should get X(s)= S-2 2)/(S-4)(s+6) for (s-4)(8+6) -). Then, find x(t)= L-(x(s))= 5 -3t (write 5/6 by 6' e^{-3t} by e and sin(2t) or cos(3t) by sin(2t) or cos(3t)).
DE Homework 7 Solve the following equations by using Laplace transform: 1. " - x'- 2.=...
DE Homework 7 Solve the following equations by using Laplace transform: 1. " - x'- 2.= 0,x(0) = 0,2'(0) = 2. 2. " + x = sin 2t, (0) = 1, x'(0) = 2. 3. " + 3x + 25 = t,x(0) = 0,2'0) = 0. 4. " + 9. = 1, 2(0) = 0,2'(0) = 0. 5. x' + 2y' + x = 0,x'- ' + y = 0,«(0) = 0, y(0) = 1. 6. " + 2x +...
Question 2: (26 marks) 2.1 Find the The Laplace transform of the following function t, if...
Question 2: (26 marks) 2.1 Find the The Laplace transform of the following function t, if 03t<1 2t, if t1 [3] 2.2 Find the inverse Laplace transform of 10e 2 52 - 53 +632 - 25 + 5 (10] 2.3 Find y(4) if y(t) = u(t){t - 2)2 - us(t)/(t - 3) - 2) - us(t)e' (51 2.4 Solve the following initial value problem given by y" + 4y = 28.(t) (0)=1/(0) = 0 181 Question 3: (17 marks) Let...
Consider the following initial value problem, representing the response of a damped oscillator subject to the discontinuous applied force f(t): y" +2y +10y = f(t), y(0) = 6, 7(0) = -3, f(t) = (1 3<t<4, 10 otherwise. {o In the following parts, use h(t -c) for the Heaviside function he(t) when necessary. a. First, compute the Laplace transform of f(t). L{f(t)}(s) = b. Next, take the Laplace transform of the left-hand-side of the differential equation, set it equal to your...
(write After use Laplace Transform to transform the following initial value problem x" + 3x' + 2x=2e-t, x(O) = x'(0)=0, you should get X(s)= S-2 fraction as (S-2)/(S-4)(s+6) for (s-4)(3+6) -). Then, find x(t) = L-2(x(s)= 5 (write 5/6 by 6 -3t e^{-3t} by e and sin(2t) or cos(3t) by sin(2t) or cos(3t)).
(write After use Laplace Transform to transform the following initial value problem x" + 2x' +x=3, x(0)=0,x'(0)=1, you should get X(s)= S-2 fraction as (S-2)/(S-4)(8+6) for -). Then, find x(t) = £-2(x(s)= (s-4)(3+6) (write 5/6 by 5 -3t 6' , e^{-3t} by e and sin(2t) or cos(3t) by sin(2t) or cos(3t)).
Solve each differential equation. (Don't use the Laplace transform. 3. IVP: y + cos(x + y) + (x – y + cos(x + y)) = 0, y(0) = 7. If the equation is exact equation, then solve it. If not, find only an exact equation.
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)...
Consider the following nonlinear differential equation, which models the unforced, undamped motion of a "soft" spring that does not obey Hooke's Law. (Here x denotes the position of a block attached to the spring, and the primes denote derivatives with respect to time t.) Note: x means x cubed notx a. Transform the second-order de. above into an equivalent system of first-order de.s. b. Use MATLAB's ode45 solver to generate a merical solution of this system aver the interval 0-t-6π...
Find the Laplace Transform for each of the following: 1. L{2sin x + 3e0s 22}= (W) *** ** (m 3 to the (s? +1)(s2 + 4) 2. 1{eosusa)= ) og 2 ( 6+23+25 ( 6+2+25 2s ZS 3. Find the inverse Laplace Transform L'{- S +1 (4) tsint (B) ’sint (0) (D) rcost
QUESTION 2 use to the following initial value problem (write fraction as (s- After Laplace Transform transform x" + 2x' +x=3, x(O)=0,x'(0)=1, you should get X(s)= S-2 2)/(S-4)(s+6) for (s-4)(8+6) -). Then, find x(t)= L-(x(s))= 5 -3t (write 5/6 by 6' e^{-3t} by e and sin(2t) or cos(3t) by sin(2t) or cos(3t)).
DE Homework 7 Solve the following equations by using Laplace transform: 1. " - x'- 2.= 0,x(0) = 0,2'(0) = 2. 2. " + x = sin 2t, (0) = 1, x'(0) = 2. 3. " + 3x + 25 = t,x(0) = 0,2'0) = 0. 4. " + 9. = 1, 2(0) = 0,2'(0) = 0. 5. x' + 2y' + x = 0,x'- ' + y = 0,«(0) = 0, y(0) = 1. 6. " + 2x +...
Question 2: (26 marks) 2.1 Find the The Laplace transform of the following function t, if 03t<1 2t, if t1 [3] 2.2 Find the inverse Laplace transform of 10e 2 52 - 53 +632 - 25 + 5 (10] 2.3 Find y(4) if y(t) = u(t){t - 2)2 - us(t)/(t - 3) - 2) - us(t)e' (51 2.4 Solve the following initial value problem given by y" + 4y = 28.(t) (0)=1/(0) = 0 181 Question 3: (17 marks) Let...
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