Explanation Definition used: Equilibrium values: If d y d x = g ( y ) is an autonomous differential equation, then the values of y for which d y d x = 0 are called equilibrium values. Calculation: The given differential equation is L d i d t + R i = V ; V , L , R > 0 . Rewrite the given equation as d i d t = V L − R L i or d i d t = R L ( V R − i ) . Consider the function u ( i ) = R L ( V R − i ) . In order to find the equilibrium points, substitute 0 for d i d t in d i d t = R L ( V R − i ) and obtain the values of i as follows. R L ( V R − i ) = 0 V R − i = 0 i = V R Thus, the equilibrium point is i = V R . Therefore, the phase line has two intervals [ 0 , V R ] and [ V R , ∞ ) . Now compute sign value for d i d t in the two intervals as shown below in Table 1. Interval Sign of d i d t [ 0 , V R ] d i d t = R L ( V R − 1 2 ⋅ V R ) = R L ( V 2 R ) = V 2 L d i d t > 0 ( ∵ V , L > 0 ) [ V R , ∞ ) d i d t = R L ( V R − 2 ⋅ V R ) = R L ( − V R ) = − V L d i d t < 0 ( ∵ V , L > 0 ) Table 1 From Table 1, it is observed that the value of d i d t = R L ( V R − i ) is negative in the interval [ V R , ∞ ) and it is positive in the interval [ 0 , V R ] . Differentiate the equation d i d t = R L ( V R − i ) with respect to t . d 2 i d t 2 = − ( R L ) d i d t = − ( R L ) ( R L ( V R − i ) ) = − ( R L ) 2 ( V R − i ) Now compute the values of i when d 2 i d t 2 = 0

Pearson eText University Calculus: Early Transcendentals -- Instant Access (Pearson+)

4th Edition

ISBN: 9780136880912

Author: Joel Hass, Christopher Heil

Publisher: PEARSON+

See similar textbooks

Videos

Question

Chapter 16.4, Problem 19E

To determine

To sketch: The solution curve using the phase line for the equation Ldidt+Ri=V and also to determine the result of the current as t→∞.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 16.4, Problem 18E

Chapter 16.4, Problem 20E

Students have asked these similar questions

Find the derivative of the function. m(t) = -4t (6t7 - 1)6

Find the derivative of the function. y= (8x²-6x²+3)4

Find f[g(x)] and g[f(x)]. f(x)=√√x+5, g(x) = 12x² -

Chapter 16 Solutions

Pearson eText University Calculus: Early Transcendentals -- Instant Access (Pearson+)

Ch. 16.1 - In Exercises 1–4, match the differential equations...Ch. 16.1 - In Exercises 1–4, match the differential equations...Ch. 16.1 - In Exercises 1–4, match the differential equations...Ch. 16.1 - In Exercises 1–4, match the differential equations...Ch. 16.1 - In Exercises 5 and 6, copy the slope fields, and...Ch. 16.1 - In Exercises 5 and 6, copy the slope fields, and...Ch. 16.1 - Prob. 7E Ch. 16.1 - Prob. 8E Ch. 16.1 - Prob. 9E Ch. 16.1 - Prob. 10E

Knowledge Booster

$Calculus$

Learn more about

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

The solution curve using the phase line for the equation L d i d t + R i = V and also to determine the result of the current as t → ∞ .

Videos

To sketch: The solution curve using the phase line for the equation Ldidt+Ri=V and also to determine the result of the current as t→∞.

Want to see the full answer?

Chapter 16 Solutions