To find the solution of the Initial-Value Problem
To find the solution of the Initial-Value Problem \(\left\{\begin{array}{l}y \prime \prime-4 y=16 \cos 2 t \\ y(0)=0 \\ y^{\prime}(0)=0\end{array}\right.\) the
Laplace Transform was applied and it was obtained as "Laplace Transform" of the unknown function \(y=f(t)\), the following:
\(L\{f(t)\}=\frac{1}{s-2}-\frac{s}{s+2}-\frac{2 s}{s^{2}+4}\)
None of them
\(L\{f(t)\}=\frac{2 s}{s-2}-\frac{1}{s+2}-\frac{s}{s^{2}+4}\)
\(L\{f(t)\}=\frac{1}{s-2}+\frac{1}{s+2}-\frac{2 s}{s^{2}+4}\)
\(L\{f(t)\}=\frac{2}{s-1}+\frac{1}{s+4}-\frac{2 s}{s^{2}+4}\)
Homework Answers
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Use the Laplace transform to solve the given initial-value problem.
Use the Laplace transform to solve the given initial-value problem.$$ y^{\prime}+y=f(t), \quad y(0)=0, \text { where } f(t)=\left\{\begin{array}{rr} 0, & 0 \leq t<1 \\ 5, & t \geq 1 \end{array}\right. $$
Question 9 3 pts The Laplace transform of the piecewise continuous function J4, 0< < 3...
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...
The objective of this question is to find the solution of the following initial-value problem using...
The objective of this question is to find the solution of the
following initial-value problem using the Laplace transform.
The objective of this question is to find the solution of the following initial-value problem using the Laplace transform y"ye2 y(0) 0 y'(0)=0 [You need to use the Laplace and the inverse Laplace transform to solve this problem. No credit will be granted for using any other technique]. Part a) (10 points) Let Y(s) = L{y(t)}, find an expression for Y(s)...
Consider the following initial value problem, representing the response of an undamped oscillator subject to the...
Consider the following initial value problem, representing the response of an undamped oscillator subject to the ramp loading applied force g(t): 0<t<2, y' + 25y = g(t), y(0) = 0, y(0) = -2, g(t) = 2<t<6, otherwise. 0 t - 2 2 2 In the following parts, use h(t – c) for the Heaviside function he(t) when necessary. a. First, compute the Laplace transform of g(t). L{f(t)}(s) b. Next, take the Laplace transform of the left-hand-side of the differential equation,...
Find the Laplace transform Y(s) = L{y} of the solution of the given initial value problem:...
Find the Laplace transform Y(s) = L{y} of the solution of the given initial value problem: 1, y' + 9 = 0<t<T 0,7 <t< y(0) = 5, y'(0) = 4
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...
Part B (5 points each] An initial value problem y' + 2y = f(©),y(0) = 0...
Part B (5 points each] An initial value problem y' + 2y = f(©),y(0) = 0 is to be solved by Laplace transforms. (B-1) When f(t) is depicted in the following, show that its Laplace transform can be obtained as f(t) 4 4e F(s) = [[f(t)) = 5ż (1-es). 4 -S s V 2 0 1 2: (B-2) Show that the Laplace transform of the solution, Y(s) = Ly(0)], can be obtained as 4 4(+ 1) Y(s) = s-(s +...
Question (2): Laplace Transforms a) Find the Laplace Transform of the following using the Laplace Transform...
Question (2): Laplace Transformsa) Find the Laplace Transform of the following using the Laplace Transform table provided in the back:$$ f(t)=\frac{1}{4}\left(3 e^{-2 t}-8 e^{-4 t}+9 e^{-6 t}\right) u(t) $$b) Find the inverse Laplace Transform \(F(s)\) of the following function \(f(t)\) using the table:$$ f(t)=\frac{12 s^{2}(s+1)}{\left(8 s^{2}+5 s+800\right)(s+5)^{2}(10 s+8)} $$
I have the first method complete, but I can't figure out thesecond method. I have...
I have the first method complete, but I can't figure out the
second method Could
someone please show how to use the second method?2. Find the unit step response of:$$ \begin{aligned} \dot{\overrightarrow{\mathbf{x}}}(t) &=\left[\begin{array}{cc} 0 & 1 \\ -2 & -2 \end{array}\right] \overrightarrow{\mathbf{x}}(t)+\left[\begin{array}{l} 1 \\ 1 \end{array}\right] u(t) \\ y(t) &=\left[\begin{array}{cc} 2 & 3 \end{array}\right] \overrightarrow{\mathbf{x}}(t) \end{aligned} $$by two methods (1): transfer function and then (2) \(y(t)=\mathbf{C} e^{\mathbf{A} t} \overrightarrow{\mathbf{x}}(0)+\mathbf{C} \int_{0}^{t} e^{\mathbf{A}(t-\tau)} \mathbf{B} u(\tau) d \tau+\mathbf{D} u(t)\). Re-member that the Laplace...
In this exercise we will use the Laplace transform to solve the following initial value problem:
In this exercise we will use the Laplace transform to solve the following initial value problem: y"-2y'+ 17y-17, y(0)=0, y'(0)=1 (1) First, using Y for the Laplace transform of y(t), i.e., Y =L(y(t)), find the equation obtained by taking the Laplace transform of the initial value problem (2) Next solve for Y= (3) Finally apply the inverse Laplace transform to find y(t)
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...
The objective of this question is to find the solution of the
following initial-value problem using the Laplace transform.
The objective of this question is to find the solution of the following initial-value problem using the Laplace transform y"ye2 y(0) 0 y'(0)=0 [You need to use the Laplace and the inverse Laplace transform to solve this problem. No credit will be granted for using any other technique]. Part a) (10 points) Let Y(s) = L{y(t)}, find an expression for Y(s)...
Consider the following initial value problem, representing the response of an undamped oscillator subject to the ramp loading applied force g(t): 0<t<2, y' + 25y = g(t), y(0) = 0, y(0) = -2, g(t) = 2<t<6, otherwise. 0 t - 2 2 2 In the following parts, use h(t – c) for the Heaviside function he(t) when necessary. a. First, compute the Laplace transform of g(t). L{f(t)}(s) b. Next, take the Laplace transform of the left-hand-side of the differential equation,...
Find the Laplace transform Y(s) = L{y} of the solution of the given initial value problem: 1, y' + 9 = 0<t<T 0,7 <t< y(0) = 5, y'(0) = 4
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...
Part B (5 points each] An initial value problem y' + 2y = f(©),y(0) = 0 is to be solved by Laplace transforms. (B-1) When f(t) is depicted in the following, show that its Laplace transform can be obtained as f(t) 4 4e F(s) = [[f(t)) = 5ż (1-es). 4 -S s V 2 0 1 2: (B-2) Show that the Laplace transform of the solution, Y(s) = Ly(0)], can be obtained as 4 4(+ 1) Y(s) = s-(s +...
Question (2): Laplace Transformsa) Find the Laplace Transform of the following using the Laplace Transform table provided in the back:$$ f(t)=\frac{1}{4}\left(3 e^{-2 t}-8 e^{-4 t}+9 e^{-6 t}\right) u(t) $$b) Find the inverse Laplace Transform \(F(s)\) of the following function \(f(t)\) using the table:$$ f(t)=\frac{12 s^{2}(s+1)}{\left(8 s^{2}+5 s+800\right)(s+5)^{2}(10 s+8)} $$
I have the first method complete, but I can't figure out the
second method Could
someone please show how to use the second method?2. Find the unit step response of:$$ \begin{aligned} \dot{\overrightarrow{\mathbf{x}}}(t) &=\left[\begin{array}{cc} 0 & 1 \\ -2 & -2 \end{array}\right] \overrightarrow{\mathbf{x}}(t)+\left[\begin{array}{l} 1 \\ 1 \end{array}\right] u(t) \\ y(t) &=\left[\begin{array}{cc} 2 & 3 \end{array}\right] \overrightarrow{\mathbf{x}}(t) \end{aligned} $$by two methods (1): transfer function and then (2) \(y(t)=\mathbf{C} e^{\mathbf{A} t} \overrightarrow{\mathbf{x}}(0)+\mathbf{C} \int_{0}^{t} e^{\mathbf{A}(t-\tau)} \mathbf{B} u(\tau) d \tau+\mathbf{D} u(t)\). Re-member that the Laplace...
In this exercise we will use the Laplace transform to solve the following initial value problem: y"-2y'+ 17y-17, y(0)=0, y'(0)=1 (1) First, using Y for the Laplace transform of y(t), i.e., Y =L(y(t)), find the equation obtained by taking the Laplace transform of the initial value problem (2) Next solve for Y= (3) Finally apply the inverse Laplace transform to find y(t)
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