Explanation Given: The electromotive force E ( t ) = { 120 , 0 ≤ t ≤ 20 0 , t > 20 is applied to a circuit with inductance 20 henries and the resistance 2 ohms. Procedure used: 1. Write the circuit equation. 2. Find the integrating factor. 3. Obtain the final solution of differential equation. 4. Substitute the condition for obtain the constant value. Calculation: Consider the LR-series, L d i d t + R i = E ( t ) (1) Here, R is resistance, L is inductance, i is current and E ( t ) is electromotive force. Case-1 : 0 ≤ t ≤ 20 Substitute 20 for L , 2 for R and 120 for E ( t ) in equation (1). 20 d i d t + 2 i = 120 d i d t + 1 10 i = 6 . (2) The solution of differential equation is, i ( I .F ) = ∫ 6 ( I .F ) d t + c (3) Obtain integrating factor of differential equation in equation (2). I .F= e ∫ 1 10 d t I .F= e 1 10 t Substitute e 1 10 t for I .F in equation (3) and integrate. i ( e 1 10 t ) = ∫ 6 ( e 1 10 t ) d t + c i ( e 1 10 t ) = 60 e 1 10 t + c i = 60 + c e − 1 10 t (4) Substitute 0 for i and 0 for t in equation (4)

6th Edition

ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Chapter 2.7, Problem 33E

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The current i(t) if i(0)=0.

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Chapter 2.7, Problem 32E

Chapter 2.7, Problem 34E

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2. Consider the ODE u' = ƒ (u) = u² + r where r is a parameter that can take the values r = −1, −0.5, -0.1, 0.1. For each value of r: (a) Sketch ƒ(u) = u² + r and determine the equilibrium points. (b) Draw the phase line. (d) Determine the stability of the equilibrium points. (d) Plot the direction field and some sample solutions,i.e., u(t) (e) Describe how location of the equilibrium points and their stability change as you increase the parameter r. (f) Using the matlab program phaseline.m generate a solution for each value of r and the initial condition u(0) = 0.9. Print and turn in your result for r = −1. Do not forget to add a figure caption. (g) In the matlab program phaseline.m set the initial condition to u(0) = 1.1 and simulate the ode over the time interval t = [0, 10] for different values of r. What happens? Why? You do not need to turn in a plot for (g), just describe what happens.

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The current i ( t ) if i ( 0 ) = 0 .

Solution Summary: The author explains how the electromotive force E(t) is applied to a circuit with inductance and resistance.

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The current i(t) if i(0)=0.

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Chapter 2 Solutions