Explanation Given data: The differential equation is, d 2 v ( t ) d t 2 + 4 d v ( t ) d t + 4 v ( t ) = 7 e − t (1) The initial condition v ( 0 ) = v ' ( 0 ) = 2 . Formula used: Write the general expression for the Laplace transform. F ( s ) = L [ f ( t ) ] (2) Write the general expression for the inverse Laplace transform. f ( t ) = L − 1 [ F ( s ) ] (3) Write the general expressions to find the Laplace transform function. L [ d f d t ] = s F ( s ) − f ( 0 − ) (4) L [ d 2 f d t 2 ] = s 2 F ( s ) − s f ( 0 − ) − f ' ( 0 − ) (5) L [ e − a t ] = 1 s + a (6) Here, s is a complex variable, s + a is the frequency shift or frequency translation. Write the general expressions to find the inverse Laplace transform function. L − 1 [ 1 s + a ] = e − a t u ( t ) (7) L − 1 [ 1 ( s + a ) 2 ] = t e − a t u ( t ) (8) Calculation: Apply Laplace transform function given in equation (2), (4), (5) and (6) to equation (1). [ s 2 V ( s ) − s v ( 0 − ) − v ' ( 0 − ) ] + 4 [ s V ( s ) − v ( 0 − ) ] + 4 V ( s ) = 7 ( 1 s + 1 ) [ s 2 V ( s ) − s v ( 0 ) − v ' ( 0 ) ] + 4 [ s V ( s ) − v ( 0 ) ] + 4 V ( s ) = 7 s + 1 { ∵ v ( 0 − ) = v ( 0 ) , v ' ( 0 − ) = v ' ( 0 ) } (9) Substitute 2 for v ( 0 ) and 2 for v ' ( 0 ) in equation (9). [ s 2 V ( s ) − s ( 2 ) − ( 2 ) ] + 4 [ s V ( s ) − ( 2 ) ] + 4 V ( s ) = 7 s + 1 s 2 V ( s ) + 4 s V ( s ) − 8 + 4 V ( s ) = 7 s + 1 + 2 s + 2 ( s 2 + 4 s + 4 ) V ( s ) = 7 s + 1 + 2 s + 2 + 8 ( s 2 + 4 s + 4 ) V ( s ) = 7 + ( 2 s + 10 ) ( s + 1 ) s + 1 (10) Rearrange the equation (10) to find V ( s ) . V ( s ) = 7 + ( 2 s + 10 ) ( s + 1 ) ( s + 1 ) ( s 2 + 4 s + 4 ) = 7 + ( 2 s 2 + 2 s + 10 s + 10 ) ( s + 1 ) ( s 2 + 4 s + 2 2 ) V ( s ) = 2 s 2 + 12 s + 17 ( s + 1 ) ( s + 2 ) 2 { ∵ a 2 + 2 a b + b 2 = ( a + b ) 2 } (11) Expand V ( s ) using partial fraction. V ( s ) = 2 s 2 + 12 s + 17 ( s + 1 ) ( s + 2 ) 2 = A s + 1 + B ( s + 2 ) 2 + C s + 2 (12) Here, A , B , and C are the constants. Now, to find the constants by using residue method. Constant A : A = ( s + 1 ) V ( s ) | s + 1 = 0 (13) Substitute equation (11) in equation (13) to find the constant A . A = ( s + 1 ) × 2 s 2 + 12 s + 17 ( s + 1 ) ( s + 2 ) 2 | s = − 1 = 2 s 2 + 12 s + 17 ( s + 2 ) 2 | s = − 1 = 2 ( − 1 ) 2 + 12 ( − 1 ) + 17 ( − 1 + 2 ) 2 = 7 Constant B : B = ( s + 2 ) 2 V ( s ) | s + 2 = 0 (14) Substitute equation (11) in equation (14) to find the constant B

6th Edition

ISBN: 8220102801448

Author: Alexander

Publisher: YUZU

See similar textbooks

Concept explainers

Transfer function

A transfer function (also known as system function or network function) of a system, subsystem, or component is a mathematical function that modifies the output of a system in each possible input. They are widely used in electronics and control systems.

Convolution Integral

Among all the electrical engineering students, this topic of convolution integral is very confusing. It is a mathematical operation of two functions f and g that produce another third type of function (f * g) , and this expresses how the shape of one is …

Question

Chapter 15.6, Problem 15PP

To determine

Find the time-varying function v(t) for the given differential equation using Laplace transform method.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 15.5, Problem 14PP

Chapter 15.6, Problem 16PP

Students have asked these similar questions

Need help for this following question

Help me out ASAP. THANK YOU!

I need a solution within 15 minutes

Chapter 15 Solutions

EBK FUNDAMENTALS OF ELECTRIC CIRCUITS

Ch. 15.2 - Prob. 1PP Ch. 15.2 - Prob. 2PP Ch. 15.3 - Prob. 3PP Ch. 15.3 - Prob. 4PP Ch. 15.3 - Prob. 5PP Ch. 15.3 - Prob. 6PP Ch. 15.3 - Obtain the initial and the final values of...Ch. 15.4 - Prob. 8PP Ch. 15.4 - Find f(t) if F(s)=48(s+2)(s+1)(s+3)(s+4)Ch. 15.4 - Obtain g(t) if G(s)=s3+2s+6s(s+1)2(s+3)

Ch. 15.4 - Find g(t) given that G(s)=20(s+1)(s2+4s+13)Ch. 15.5 - Graphically convolve the two functions in Fig....Ch. 15.5 - Given g(t) and f(t) in Fig. 15.20, graphically...Ch. 15.5 - Use convolution to find vo(t) in the circuit of...Ch. 15.6 - Prob. 15PP Ch. 15.6 - Prob. 16PP Ch. 15 - Prob. 1RQ Ch. 15 - Prob. 2RQ Ch. 15 - Prob. 3RQ Ch. 15 - Prob. 4RQ Ch. 15 - Prob. 5RQ Ch. 15 - If F(s) = 1/(s + 2), then f(t) is (a) e2t u(t) (b)...Ch. 15 - Prob. 7RQ Ch. 15 - Prob. 8RQ Ch. 15 - Prob. 9RQ Ch. 15 - Prob. 10RQ Ch. 15 - Prob. 1P Ch. 15 - Prob. 2P Ch. 15 - Prob. 3P Ch. 15 - Prob. 4P Ch. 15 - Prob. 5P Ch. 15 - Prob. 6P Ch. 15 - Prob. 7P Ch. 15 - Prob. 8P Ch. 15 - Prob. 9P Ch. 15 - Prob. 10P Ch. 15 - Find F(s) if: (a) ft=6etcosh2t (b) ft=3te2tsinh4t...Ch. 15 - If g(t) = 4e 2t cos 4t, find G(s).Ch. 15 - Prob. 13P Ch. 15 - Prob. 14P Ch. 15 - Prob. 15P Ch. 15 - Prob. 16P Ch. 15 - Prob. 17P Ch. 15 - Prob. 18P Ch. 15 - Prob. 19P Ch. 15 - Prob. 20P Ch. 15 - Prob. 21P Ch. 15 - Prob. 22P Ch. 15 - Prob. 23P Ch. 15 - Design a problem to help other students better...Ch. 15 - Let F(s)=18(s+1)(s+2)(s+3) (a) Use the initial and...Ch. 15 - Determine the initial and final values of f(t), if...Ch. 15 - Prob. 27P Ch. 15 - Prob. 28P Ch. 15 - Prob. 29P Ch. 15 - Prob. 30P Ch. 15 - Find f(t) for each F(s): (a) 10ss+1s+2s+3 (b)...Ch. 15 - Prob. 32P Ch. 15 - Prob. 33P Ch. 15 - Prob. 34P Ch. 15 - Obtain f(t) for the following transforms: (a)...Ch. 15 - Prob. 36P Ch. 15 - Prob. 37P Ch. 15 - Prob. 38P Ch. 15 - Determine f(t) if: (a)...Ch. 15 - Show that...Ch. 15 - Prob. 41P Ch. 15 - Design a problem to help other students better...Ch. 15 - Prob. 43P Ch. 15 - Prob. 44P Ch. 15 - Given h(t) = 4e2tu(t) and x(t) = (t) 2e 2tu(t),...Ch. 15 - Given the following functions...Ch. 15 - A system has the transfer function...Ch. 15 - Find f(t) using convolution given that: (a)...Ch. 15 - Prob. 49P Ch. 15 - Prob. 50P Ch. 15 - Given that v(0) = 5 and dv(0)/dt = 10, solve...Ch. 15 - Prob. 52P Ch. 15 - Prob. 53P Ch. 15 - Design a problem to help other students better...Ch. 15 - Prob. 55P Ch. 15 - Solve for v(t) in the integrodifferential equation...Ch. 15 - Prob. 57P Ch. 15 - Given that dvdt+2v+50tv()d=4u(t) with v(0) = 1,...Ch. 15 - Solve the integrodifferential equation...Ch. 15 - Prob. 60P

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, electrical-engineering and related others by exploring similar questions and additional content below.

Similar questions

SEE MORE QUESTIONS

Recommended textbooks for you

Introductory Circuit Analysis (13th Edition)
Electrical Engineering
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:PEARSON
Delmar's Standard Textbook Of Electricity
Electrical Engineering
ISBN:9781337900348
Author:Stephen L. Herman
Publisher:Cengage Learning
Programmable Logic Controllers
Electrical Engineering
ISBN:9780073373843
Author:Frank D. Petruzella
Publisher:McGraw-Hill Education
Fundamentals of Electric Circuits
Electrical Engineering
ISBN:9780078028229
Author:Charles K Alexander, Matthew Sadiku
Publisher:McGraw-Hill Education
Electric Circuits. (11th Edition)
Electrical Engineering
ISBN:9780134746968
Author:James W. Nilsson, Susan Riedel
Publisher:PEARSON
Engineering Electromagnetics
Electrical Engineering
ISBN:9780078028151
Author:Hayt, William H. (william Hart), Jr, BUCK, John A.
Publisher:Mcgraw-hill Education,