(a) Explanation Concept: 1. A piecewise defined continuous function of the form f ( t ) = { g ( t ) , 0 ≤ t < a h ( t ) , t ≥ a can be expressed as f ( t ) = g ( t ) − g ( t ) U ( t − a ) + h ( t ) U ( t − a ) . If F ( s ) = L { f ( t ) } and a > 0 , then L { f ( t − a ) U ( t − a ) } = e − a s F ( s ) L { U ( t − a ) } = e − a s s 2. The Laplace transformation of a linear differential equation with constant coefficient can be expressed as, a n L { d n y d t n } = a n [ s n Y ( s ) − s ( n − 1 ) y ( 0 ) − ⋯ − y ( n − 1 ) ( 0 ) ] 3. The formula for the Laplace transform is as follows, L { 1 } = 1 s L { sin k t } = k s 2 + k 2 L { cos k t } = s s 2 + k 2 4. The formula for the inverse Laplace transform is as follows, L − 1 { 1 s − a } = e a t 5. If F ( s ) = L { f ( t ) } and a > 0 , then L − 1 { F ( s − a ) } = e a t f ( t ) 6. The linearity property of the Laplace transformation is as below. If a , b be any constant and f , g be any function of t , then L { a f ( t ) + b g ( t ) } = a L { f ( t ) } + b L { g ( t ) } Calculation: The expression for the linear differential L R series circuit is as follows: L d i d t + R i = E ( t ) Substitute the value of R and L in the above expression. d i d t + 10 i = E ( t ) The step function for the given graph is as follows: E ( t ) = { sin t , 0 ≤ t < 3 π 2 0 , t ≥ 3 π 2 Use the definition 1 of Laplace transforms to write the unit step function for the function E ( t ) . E ( t ) = sin t − sin t U ( t − 3 π 2 ) E ( t ) = sin t − cos ( t − 3 π 2 ) U ( t − 3 π 2 ) Use the definition 2 of Laplace transforms to write the Laplace transform of the above linear differential L R series circuit and substitute E ( t ) = sin t − cos ( t − 3 π 2 ) U ( t − 3 π 2 ) , then equation becomes s L { i } − i ( 0 ) + 10 L { i } = L { sin t − cos ( t − 3 π 2 ) U ( t − 3 π 2 ) } … … . ( 1 ) Substitute the initial given value i ( 0 ) = 0 in equation (1) and solve. ( s + 10 ) L { i } = L { sin t } − L { cos ( t − 3 π 2 ) U ( t − 3 π 2 ) } ( s + 10 ) L { i } = 1 s 2 + 1 − e − 3 π s 2 s s 2 + 1 L { i } = 1 ( s 2 + 1 ) ( s + 10 ) − e − 3 π s 2 s ( s 2 + 1 ) ( s + 10 ) From the above expression, the result obtained is as follows: L { i } = 1 ( s 2 + 1 ) ( s + 10 ) − e − 3 π s 2 s ( s 2 + 1 ) ( s + 10 ) … … ( 2 ) The term 1 ( s 2 + 1 ) ( s + 10 ) in equation (2) is simplify as follows using partial fraction, 1 ( s + 10 ) ( s 2 + 1 ) = a 1 s + a 0 s 2 + 1 + a 2 s + 10 … … ( 3 ) Multiply with ( s + 10 ) ( s 2 + 1 ) on both side of above expression. 1 ⋅ ( s + 10 ) ( s 2 + 1 ) ( s + 10 ) ( s 2 + 1 ) = ( a 1 s + a 0 ) ( s + 10 ) ( s 2 + 1 ) s 2 + 1 + a 2 ( s + 10 ) ( s 2 + 1 ) s + 10 1 = ( a 1 s + a 0 ) ( s + 10 ) + a 2 ( s 2 + 1 ) From the above expression, the result obtained is as follows: 1 = ( a 1 s + a 0 ) ( s + 10 ) + a 2 ( s 2 + 1 ) … … ( 4 ) Substitute s = − 10 in equation (4). 1 = ( a 1 ( − 10 ) + a 0 ) ( ( − 10 ) + 10 ) + a 2 ( ( − 10 ) 2 + 1 ) 1 = 101 a 2 a 2 = 1 101 Substitute the value of a 2 = 1 101 in equation (4) and solve. 1 = ( a 1 s + a 0 ) ( s + 10 ) + 1 101 ( s 2 + 1 ) 1 = a 1 s 2 + 10 a 1 s + a 0 s + 10 a 0 + 1 101 s 2 + 1 101 1 = s 2 ( a 1 + 1 101 ) + s ( a 0 + 10 a 1 ) + ( 10 a 0 + 1 101 ) Compare the similar coefficients of on both sides of the above expression. 10 a 0 + 1 101 = 1 10 a 1 + a 0 = 0 a 1 + 1 101 = 0 Solve the above system of equation. a 0 = 10 101 a 1 = − 1 101 Substitute the value of a 0 , a 1 and a 2 in equation (3) (b) To determine To draw: The graph for the current i ( t ) for 0 ≤ t ≤ 6 , and then find the i max and i min from the graph.

Differential Equations with Boundary-Value Problems (MindTap Course List)

9th Edition

ISBN: 9781305965799

Author: Dennis G. Zill

Publisher: Cengage Learning

See similar textbooks

Videos

Question

Chapter 7.3, Problem 75E

(a)

To determine

The expression for the current i(t) from the linear differential equation of LR series circuit using Laplace transform.

(b)

To determine

To draw: The graph for the current i(t) for 0≤t≤6, and then find the imax and imin from the graph.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 7.3, Problem 74E

Chapter 7.3, Problem 76E

Students have asked these similar questions

2. Hypothesis Testing - Two Sample Means A nutritionist is investigating the effect of two different diet programs, A and B, on weight loss. Two independent samples of adults were randomly assigned to each diet for 12 weeks. The weight losses (in kg) are normally distributed. Sample A: n = 35, 4.8, s = 1.2 Sample B: n=40, 4.3, 8 = 1.0 Questions: a) State the null and alternative hypotheses to test whether there is a significant difference in mean weight loss between the two diet programs. b) Perform a hypothesis test at the 5% significance level and interpret the result. c) Compute a 95% confidence interval for the difference in means and interpret it. d) Discuss assumptions of this test and explain how violations of these assumptions could impact the results.

1. Sampling Distribution and the Central Limit Theorem A company produces batteries with a mean lifetime of 300 hours and a standard deviation of 50 hours. The lifetimes are not normally distributed—they are right-skewed due to some batteries lasting unusually long. Suppose a quality control analyst selects a random sample of 64 batteries from a large production batch. Questions: a) Explain whether the distribution of sample means will be approximately normal. Justify your answer using the Central Limit Theorem. b) Compute the mean and standard deviation of the sampling distribution of the sample mean. c) What is the probability that the sample mean lifetime of the 64 batteries exceeds 310 hours? d) Discuss how the sample size affects the shape and variability of the sampling distribution.

An airplane flies due west at an airspeed of 428 mph. The wind blows in the direction of 41° south of west at 50 mph. What is the ground speed of the airplane? What is the bearing of the airplane? 428 mph 41° 50 mph a. The ground speed of the airplane is b. The bearing of the airplane is mph. south of west.

Chapter 7 Solutions

Differential Equations with Boundary-Value Problems (MindTap Course List)

Ch. 7.1 - In Problems 118 use Definition 7.1.1 to find f(t)....Ch. 7.1 - In Problems 118 use Definition 7.1.1 to find f(t)....Ch. 7.1 - In Problems 118 use Definition 7.1.1 to find f(t)....Ch. 7.1 - Prob. 4E Ch. 7.1 - Prob. 5E Ch. 7.1 - Prob. 6E Ch. 7.1 - Prob. 7E Ch. 7.1 - In Problems 118 use Definition 7.1.1 to find f(t)....Ch. 7.1 - In Problems 118 use Definition 7.1.1 to find f(t)....Ch. 7.1 - Prob. 10E

Knowledge Booster

Learn more about

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

The expression for the current i ( t ) from the linear differential equation of L R series circuit using Laplace transform.

Solution Summary: The author describes the expression for the current i(t) from the linear differential equation using Laplace transform.

Videos

The expression for the current i(t) from the linear differential equation of LR series circuit using Laplace transform.

To draw: The graph for the current i(t) for 0≤t≤6, and then find the imax and imin from the graph.

Want to see the full answer?

Chapter 7 Solutions