6669
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2011-02-01T00:00:01
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<p>V represents the voltage drop across the parallel combination of the capacitor (C) and resistor (R). Find the differential equation that V obeys in terms of $V_0$ and its derivatives.<br></br>
<mdo:image alt="" height="284" src="Problem3.jpg" width="513"></mdo:image><br></br>
Make up some realistic constants for a capacitor. Can you solve the resulting differential equation?<br></br>
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<mdo:image height="284" width="513" alt="" src="Problem3%27.jpg"></mdo:image><br></br>
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<br></br>
Kirchoff's Voltage Law:<br></br>
<br></br>
$V_0 = V_L + V_R + V = L\frac{dI_1}{dt} + IR + V$<br></br>
<br></br>
Current conservation at node X:<br></br>
<br></br>
$I_1 = I_2 + I_3 $<br></br>
<br></br>
$I_2 = I_C = C\frac{dV}{dt}$<br></br>
$I_3 = \frac{V}{R}$<br></br>
<br></br>
$I_1 = C\frac{dV}{dt} + \frac{V}{R} $<br></br>
<br></br>
$\frac{dI_1}{dt} = C\frac{d^2V}{dt^2} +\frac{1}{R}
\frac{dV}{dt}$<br></br>
<br></br>
$V_0 = L(C\frac{d^2V}{dt^2} +\frac{1}{R} \frac{dV}{dt}) +
(C\frac{dV}{dt} + \frac{V}{R})R + V = LC \frac{d^2V}{dt^2} +
(\frac{L}{R} + CR)\frac{dV}{dt} + 2V$<br></br>
<br></br>
The governing differential equation is:<br></br>
<br></br>
$LC \frac{d^2V}{dt^2} + (\frac{L}{R} + CR)\frac{dV}{dt} + 2V = V_0
$<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0">V = L$\frac{dI}{dt}$<br></br>
<br></br>
I = C$\frac{dV}{dt}$<br></br>
<br></br>
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Differential electricity
As a capacitor discharges, its charge changes continuously. Find
the differential equation governing this variation.
Applications
engineering
Pre-Calculus and Calculus
1st and 2nd order linear differential equations with constant coefficients
Applications
engineering
Admin
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