Solve the following differential equations subject to the specified initial conditions
- (a) d2v/dt2 + 4v = 12, v(0) = 0, dv(0)/dt = 2
- (b) d2i/dt2 + 5 di/dt + 4i = 8, i(0) = −1, di(0)/dt = 0
- (c) d2v/dt2 + 2 dv/dt + v = 3, v(0) = 5, dv(0)/dt = 1
- (d) d2i/dt2 + 2 di/dt + 5i = 10, i(0) = 4, di(0)/dt = −2
(a)
Find the expression of
Answer to Problem 29P
The expression of
Explanation of Solution
Given data:
The differential equation is,
The values of initial conditions are,
Formula used:
Write an expression to find the voltage response with the step input, if the roots of characteristic equation are real and imaginary.
Here,
Write a general expression for the roots of characteristic equation when the roots are real and imaginary.
Write an expression to solve quadratic equation.
Here,
Calculation:
From equation (1), the characteristic equation is written as follows.
From the equation (6),
Substitute
Simplify the equation.
Therefore, the roots of characteristic equations are real and imaginary.
Compare the values of roots with the equation (3) and equation (4).
Substitute
At
Therefore,
Substitute
For the step response
Substitute
Substitute
Substitute
Substitute
Substitute
Differentiate equation (11) with respect to
Substitute
Substitute
Substitute
Conclusion:
Thus, the expression of
(b)
Find the expression of
Answer to Problem 29P
The expression of
Explanation of Solution
Given data:
The differential equation is,
The values of initial conditions are,
Formula used:
Write a general expression for the current response with the step input, if the roots of characteristic equation are real and negative.
Here,
Calculation:
From equation (14), the characteristic equation is written as follows.
Compare the equation (16) with the quadratic equation
Substitute
Simplify the equation.
Therefore, the roots of characteristic equations are real and negative.
Substitute
At
Therefore,
Substitute
For the step response
Substitute
Substitute
Substitute
Substitute
Simplify the above equation to find
Differentiate equation (19) with respect to
Substitute
Substitute
Substitute equation (20) in above equation to find
Simplify the above equation to find
Substitute
Substitute
Conclusion:
Thus, the expression of
(c)
Find the expression of
Answer to Problem 29P
The expression of
Explanation of Solution
Given data:
The differential equation is,
The values of initial conditions are,
Formula used:
Write an expression to find the voltage response with the step input, if the roots of characteristic equation are real and equal.
Here,
Write a general expression for the roots of characteristic equation when the roots are real and equal.
Calculation:
From equation (21), the characteristic equation is written as follows.
Compare the equation (24) with the quadratic equation
Substitute
Simplify the equation.
Therefore, the roots of characteristic equations are real and imaginary.
Substitute
Substitute
Substitute
For the step response,
Substitute
Substitute
Substitute
Substitute
Simplify the above equation to find
Expand the equation (27) as follows:
Differentiate the above with respect to
Substitute
Substitute
Simplify the above equation to find
Substitute
Substitute
Conclusion:
Thus, the expression of
(d)
Find the expression of
Answer to Problem 29P
The expression of
Explanation of Solution
Given data:
The differential equation is,
The values of initial conditions are,
Formula used:
Write an expression to find the current response with the step input, if the roots of characteristic equation are real and imaginary.
Calculation:
From equation (28), the characteristic equation is written as follows.
From the equation (6),
Substitute
Simplify the equation.
Therefore, the roots of characteristic equations are real and imaginary.
Compare the values of roots with the equation (3) and equation (4).
Substitute
Substitute
For the step response,
Substitute
Substitute
Substitute
Substitute
Simplify the above equation to find
Substitute
Differentiate equation (34) with respect to
Substitute
Substitute
Substitute
Conclusion:
Thus, the expression of
Want to see more full solutions like this?
Chapter 8 Solutions
Fundamentals of Electric Circuits
- For a certain source-free parallel RLC circuit, R = 1 k , C = 3 μF, and L is such that the circuit response is overdamped. (a) Determine the value of L. (b) Write the equation for the voltage v across the resistor if it is known that v(0−) = 9 V and dv/dt|t=0+ = 2 V/s.arrow_forwardGenerate at random simple graphs with 10 vertices. Stop when you have constructed one with an Euler circuit. Display the Euler circuit. (Hint: use Matlab function graph) Please do not copy any existing responses. Respond in 1 hour to receive upvote.arrow_forward2. Find vo(t) in the circuit shown in Figure 2 by using Laplace Transformation and node currents while ideal op amp operates within its linear range and vg = 24u(t) mV. 20 nF HE 10 kΩ 20 nF 20 k12 HE + + + v.(t) Figure 2arrow_forward
- The switch in the circuit shown has been in position x for a long time. At t=0, the switch moves instantaneously to position y. 1. a) Construct an s-domain circuit for t>0. 2. b) Find Vo. 3. c) Find vo.arrow_forward5) Consider the circuit shown below, in which the switch opens at t=0. Find expressions for v(t), iR(t), and iL(t) for t>0. Assume that iL(t) is zero before the switch opens.arrow_forwardFor the RLC circuit given in the figure, we will find state variables such that the state equations of the system x=Ax+Bu , y=cx+du can be written in the form . Note: Accept the system output as ic.arrow_forward
- R =73 Find the complete response v0(t) and i0(t) of the circuit shown below.arrow_forwardA combinational circuit is defined by the function F1= ∑ m (3,5,7), F2 = ∑ m (4,5,7). Implement the circuit using a PLA which consists of 3 inputs (A, B and C), 3 product terms and two outputsarrow_forwardAt time t = 0 because of closing a switch, an RL circuit in series has resistance R = 50Ω andinductance L = 10H with a constant supply of V = 100V. It is desired to find the current at 0.5seconds, after finding the differential equation for the current.arrow_forward
- Consider the circuit below, where A=400 mF. Obtain an expression for iL(t) valid for all t > 0. Please show all work and step-by-step in-depth explanations.arrow_forwardThe switch in the circuit shown has been in position a for a long time. At t=0, the switch moves instantaneously to position b. 1. Derive the integrodifferential equation that governs the behavior of the voltage vo for t≥0+. 2. Show that Vo(s)=Vdc[s+(R/L)][s2+(R/L)s+(1/LC)].arrow_forward4. For the following circuit: a) find the differential equation for v; b) find the forced response of v, if the current source is I = 2 sin 2t (A). Please draw the final circuit formarrow_forward
- Introductory Circuit Analysis (13th Edition)Electrical EngineeringISBN:9780133923605Author:Robert L. BoylestadPublisher:PEARSONDelmar's Standard Textbook Of ElectricityElectrical EngineeringISBN:9781337900348Author:Stephen L. HermanPublisher:Cengage LearningProgrammable Logic ControllersElectrical EngineeringISBN:9780073373843Author:Frank D. PetruzellaPublisher:McGraw-Hill Education
- Fundamentals of Electric CircuitsElectrical EngineeringISBN:9780078028229Author:Charles K Alexander, Matthew SadikuPublisher:McGraw-Hill EducationElectric Circuits. (11th Edition)Electrical EngineeringISBN:9780134746968Author:James W. Nilsson, Susan RiedelPublisher:PEARSONEngineering ElectromagneticsElectrical EngineeringISBN:9780078028151Author:Hayt, William H. (william Hart), Jr, BUCK, John A.Publisher:Mcgraw-hill Education,