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# Name: • I will be checking for organization, conceptual understanding, and proper mathematical communication, as well as completion of the problems. • Show as much work as you can, draw sketches if necessary and clearly explain why you are doing what you are doing. • Use correct mathematical notation. • You may work with your classmates. However, please submit your own work! 1. The Series RLC Circuit We can model a flow of electric current in the RLC circuit (see the figure below) via a second-order linear differential equation with constant coefficients. − V (t) + R L C Figure The series RLC circuit. The current j (in A – amperes), is a function of time t. The resistance R (in Ω – ohms), the inductance L (in H – henrys), and the capacitance C (in F – farads) are all positive and are assumed to be known constants. The applied voltage V (in V – volts) is a given function of time. Another physical quantity that enters the discussion is the total charge q (in C – coulombs) on the capacitor at time t. The relation between charge q and current j is given by j = dq dt . (1) The flow of current in the circuit is governed by Kirchhoff ’s Voltage Law: In a closed circuit, the applied voltage is equal to the algebraic sum of the voltages across the elements in the rest of the circuit. According to the elementary electric circuit theory, we know that • The voltage across the inductor is L dj dt. • The voltage across the resistor is Rj. • The voltage across the capacitor is 1 C q. Hence, by Kirchhoff ’s Voltage Law stated above, we have the following: L dj dt + Rj + 1 C q = V (t). (2) Substituting for j from (1) into (2), we obtain the following second-order equation for q: L d 2 q dt2 + R dq dt + 1 C q = V (t) . (3) (Can you see that!? You DO NOT need to check this.) 1 Now, consider the RLC circuit with: • R = 5 Ω, • L = 1 H, • C = 0.25 F, • the input voltage V (t) = 5 cos(3t) Volts. (a) (8 points) Assuming that q(0) = q 0 (0) = 0, find the solution of the IVP. Use the method of complexification (Section 4.2) when finding qp(t). (b) (1 point) Determine the amount of charge on the capacitor at t = 2. Make sure to include units! Round your answer reasonably. (c) (1 point) Determine the current flowing in the circuit at t = 2. [Hint: Equation (1).] Make sure to include units! Round your answer reasonably. 2. Laplace Transform (10 points) Solve the following IVP. Recall that ua(t) is used to denote the Heaviside function (introduced in Section 6.2). Show all your work. dy dt − 2y = u6(t), y(0) = 3 2

Name:
• I will be checking for organization, conceptual understanding, and proper mathematical communication, as
well as completion of the problems.
• Show as much work as you can, draw sketches if necessary and clearly explain why you are doing what you are
doing.
• Use correct mathematical notation.
1. The Series RLC Circuit We can model a flow of electric current in the RLC circuit (see the figure
below) via a second-order linear differential equation with constant coefficients.

V (t) +
R L
C
Figure The series RLC circuit.
The current j (in A – amperes), is a function of time t. The resistance R (in Ω – ohms), the inductance
L (in H – henrys), and the capacitance C (in F – farads) are all positive and are assumed to be known
constants. The applied voltage V (in V – volts) is a given function of time. Another physical quantity
that enters the discussion is the total charge q (in C – coulombs) on the capacitor at time t. The relation
between charge q and current j is given by
j =
dq
dt . (1)
The flow of current in the circuit is governed by Kirchhoff ’s Voltage Law: In a closed circuit, the
applied voltage is equal to the algebraic sum of the voltages across the elements in the rest of the circuit.
According to the elementary electric circuit theory, we know that
• The voltage across the inductor is L
dj
dt.
• The voltage across the resistor is Rj.
• The voltage across the capacitor is 1
C
q.
Hence, by Kirchhoff ’s Voltage Law stated above, we have the following:
L
dj
dt + Rj +
1
C
q = V (t). (2)
Substituting for j from (1) into (2), we obtain the following second-order equation for q:
L
d
2
q
dt2
+ R
dq
dt +
1
C
q = V (t) . (3)
(Can you see that!? You DO NOT need to check this.)
1
Now, consider the RLC circuit with:
• R = 5 Ω,
• L = 1 H,
• C = 0.25 F,
• the input voltage V (t) = 5 cos(3t) Volts.
(a) (8 points) Assuming that q(0) = q
(0) = 0, find the solution of the IVP. Use the method of
complexification (Section 4.2) when finding qp(t).
(b) (1 point) Determine the amount of charge on the capacitor at t = 2. Make sure to include
(c) (1 point) Determine the current flowing in the circuit at t = 2. [Hint: Equation (1).] Make sure
2. Laplace Transform (10 points) Solve the following IVP. Recall that ua(t) is used to denote the Heaviside
function (introduced in Section 6.2). Show all your work.
dy
dt − 2y = u6(t), y(0) = 3
2

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