Explanation Theorem used: Laplace transform of a periodic function: “If f ( t ) is a piecewise continuous function on [ 0 , ∞ ) , of exponential order and periodic with period T , then L { f ( t ) } = 1 1 − e − s T ∫ 0 T e − s t f ( t ) d t Second translation or shifting property: If L − 1 { F ( s ) } = f ( t ) then, L − 1 { e − a p F ( s ) } = f ( t − a ) u ( t − a ) Formula used: (1) If a L [ x ″ ( t ) ] + b L [ x ′ ( t ) ] + c L [ x ( t ) ] = L { f ( t ) } then, a { s 2 L [ x ( t ) ] − s x ( 0 ) − x ′ ( 0 ) } + b { s L [ x ( t ) ] − x ( 0 ) } + c L [ x ( t ) ] = L { f ( t ) } (2) ( 1 + e − x t ) − 1 = 1 − e − x t + e − 2 x t − e − 3 x t + ⋯ (3) L − 1 [ 1 s ] = 1 (4) L − 1 [ ( s + a ) ( s + a ) 2 + b 2 ] = e − a t cos b t (5) L − 1 [ 1 ( s + a ) 2 + b 2 ] = e − a t 1 b sin b t Calculation: Consider the mass system with damping m d 2 x d t 2 + β d x d t + k x = f ( t ) with m = 1 2 , β = 1 , k = 5 . The above equation can be rewritten as follows. Here, f ( t ) is the meander function with amplitude 10. ( 1 2 ) x ″ + ( 1 ) x ′ + 5 x = 10 f ( t ) x ″ + 2 x ′ + 10 x = 20 f ( t ) Define the function f ( t ) as follows. It is given that f ( t ) is the meander function with time period T = 2 a with a = π , 0 ≤ t ≤ 2 π , is defined by, f ( t ) = { 1 , 0 < t < a − 1 , a < t < 2 a And outside the interval by f ( t + 2 π ) = f ( t ) . Take Laplace transform for the function f ( t ) as follows. L { f ( t ) } = 1 1 − e − s ( 2 a ) ∫ 0 2 a e − s t f ( t ) d t = 1 1 − e − 2 a s [ ∫ 0 a e − s t ( 1 ) d t + ∫ a 2 a e − s t ( − 1 ) d t ] = 1 1 − e − 2 a s [ ( e − s t − s ) 0 a − ( e − s t − s ) a 2 a ] = 1 1 − e − 2 a s [ ( − e − a s + 1 s ) + ( e − 2 a s − e − a s s ) ] The above expression can be further simplified as follows. L { f ( t ) } = 1 s ( 1 − e − 2 a s ) ( 1 − 2 e − a s + e − 2 a s ) = 1 s ( 1 + e − a s ) ( 1 − e − a s ) ( 1 − e − a s ) 2 = 1 − e − a s s ( 1 + e − a s ) = 1 − e − π s s ( 1 + e − π s ) [ ∵ a = π ] It is given that x ( 0 ) = 0 , x ′ ( 0 ) = 0 . Now take the Laplace transform for the differential equation as follows. L [ x ″ ] + 2 L [ x ′ ] + 10 L [ x ] = 20 L [ f ( t ) ] s 2 L [ x ( t ) ] − s x ( 0 ) − x ′ ( 0 ) + 2 s L [ x ( t ) ] − 2 x ( 0 ) + 10 L [ x ( t ) ] = 20 [ 1 − e − π s s ( 1 + e − π s ) ] ( s 2 + 2 s + 10 ) L [ x ( t ) ] = 20 s ( 1 − e − π s ) ( 1 + e − π s ) − 1 L [ x ( t ) ] = 20 s ( s 2 + 2 s + 10 ) [ ( 1 − e − π s ) ( 1 − e − π s + e − 2 π s − e − 3 π s + ⋯ ) ] Simplify the above expression further as shown below

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 7.4, Problem 61E

To determine

To sketch: The graph of x(t) for the indicated values of t using a graphing utility.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 7.4, Problem 60E

Chapter 7.4, Problem 62E

Students have asked these similar questions

At a Noodles & Company restaurant, the probability that a customer will order a nonalcoholic beverage is 0.49. Find the probability that in a sample of 13 customers, at least 7 will order a nonalcoholic beverage

10. In the general single period market model with = {W1, W2, W3}, one risky asset, S, and a money market account, we have So = 4 for the risky asset. Moreover, the effective rate of interest on the money market account is 5% and at time t = 1 we have W1 W2 W3 S₁ 100 50 40 21 21 21 (a) Calculate all risk-neutral probability measures for this model. [4 Marks] (b) State if the model is arbitrage-free. Give a brief reason for your answer. [2 Marks] (c) A large bank has designed an investment product with payoff X at time t = 1. Given W₁ W2 W3 X 0 1 1.5 show that X is an attainable contingent claim. [4 marks]

Question 1. (10 points) A researcher is studying tumours in mice. The growth rate for the volume of the tumour V(t) in cm³ is given by dV = 1.45V(2 In(V+1)). dt (a) (4 pts) Find all the equilibria and determine their stability using the stability condition. (b) (2 pts) Draw the phase plot f(V) versus V where f(V) = V'. You may find it helpful to use Desmos or Wolfram Alpha to plot the graph of f(V) versus V (both are free to use online), or you can plot it by hand if you like. On the plot identify each equilibrium as stable or unstable. (c) (4 pts) Draw direction arrows for the case where the tumour starts at size 3cm³ and for the case where the tumour starts at size 9cm³. Explain in biological terms what happens to the size of each of these tumours at time progresses.

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 graph of x ( t ) for the indicated values of t using a graphing utility.

Solution Summary: The author sketched the graph of x(t) for the indicated values of t using a graphing utility.

Videos

To sketch: The graph of x(t) for the indicated values of t using a graphing utility.

Want to see the full answer?

Chapter 7 Solutions