An electric circuit, consisting of a capacitor, resistor, and an electromotive force can be modeled by the differential equation dq 1 R- +で c9 = E(t), dt where R and C are constants (resistance and capacitance) and q(t) is the amount of charge on the capacitor at time t. For simplicity in the following analysis, let R = C = 1, forming the differential equation dq/dt +q = E (t). In Exer- cises 17–20, an electromotive force is given in piecewise form, a favorite among engineers. Assume that the initial charge on the capacitor is zero [q(0) = 0]. (i) Use a numerical solver to draw a graph of the charge on the capacitor during the time interval [0, 4]. (ii) Find an explicit solution and use the formula to determine the charge on the capacitor at the end of the four-second time period. 9 = 2t, if 0 < t < 2, |0, if t > 2 19. Е(()
An electric circuit, consisting of a capacitor, resistor, and an electromotive force can be modeled by the differential equation dq 1 R- +で c9 = E(t), dt where R and C are constants (resistance and capacitance) and q(t) is the amount of charge on the capacitor at time t. For simplicity in the following analysis, let R = C = 1, forming the differential equation dq/dt +q = E (t). In Exer- cises 17–20, an electromotive force is given in piecewise form, a favorite among engineers. Assume that the initial charge on the capacitor is zero [q(0) = 0]. (i) Use a numerical solver to draw a graph of the charge on the capacitor during the time interval [0, 4]. (ii) Find an explicit solution and use the formula to determine the charge on the capacitor at the end of the four-second time period. 9 = 2t, if 0 < t < 2, |0, if t > 2 19. Е(()
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![An electric circuit, consisting of a capacitor, resistor, and
an electromotive force can be modeled by the differential
equation
bp
R
1
+
dt
c9 = E(t),
where R and C are constants (resistance and capacitance) and
q = q(t) is the amount of charge on the capacitor at time
t. For simplicity in the following analysis, let R = C = 1,
forming the differential equation dq/dt + q
cises 17–20, an electromotive force is given in piecewise form,
a favorite among engineers. Assume that the initial charge on
the capacitor is zero [q(0) = 0].
(i) Use a numerical solver to draw a graph of the charge on
the capacitor during the time interval [0, 4].
(ii) Find an explicit solution and use the formula to determine
the charge on the capacitor at the end of the four-second
time period.
%3D
= E(t). In Exer-
%3D
2t, if 0 < t < 2,
19. Е(()
| 0,
if t > 2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1b1372ae-e668-4971-89ee-29da7ac7c466%2F8ae913fa-8486-4640-bbdc-d2bd237b3a6a%2F65zpfk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:An electric circuit, consisting of a capacitor, resistor, and
an electromotive force can be modeled by the differential
equation
bp
R
1
+
dt
c9 = E(t),
where R and C are constants (resistance and capacitance) and
q = q(t) is the amount of charge on the capacitor at time
t. For simplicity in the following analysis, let R = C = 1,
forming the differential equation dq/dt + q
cises 17–20, an electromotive force is given in piecewise form,
a favorite among engineers. Assume that the initial charge on
the capacitor is zero [q(0) = 0].
(i) Use a numerical solver to draw a graph of the charge on
the capacitor during the time interval [0, 4].
(ii) Find an explicit solution and use the formula to determine
the charge on the capacitor at the end of the four-second
time period.
%3D
= E(t). In Exer-
%3D
2t, if 0 < t < 2,
19. Е(()
| 0,
if t > 2
Transcribed Image Text:Solu tion:
19. The computed solution is shown in the following fig-
ure.
3f
2-
2
3
The exact solution is
|2(t – 1+e-'), if 0 <t < 2,
|2(1+e-2)e²-t, if t > 2
q(t) =
Hence q(4) = 2(1+e-2)e-2 × 0.3073.
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