User:Egm4313.s12.team4.friedman/R1

= Problem R1.4: Some Title =

Problem Statement
Derive equations (1.4.2) and (1.4.3) from (1.4.1).


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$$\displaystyle V=LC \frac{d^2 v_C}{dt^2} + RC \frac{d v_C}{dt} + v_C$$
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 * (1.4.1)
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$$\displaystyle L I'' + R I' + \frac{1}{C} I = V'$$
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 * (1.4.2)
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$$\displaystyle L Q'' + R Q' + \frac{1}{C} Q = V$$
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 * (1.4.3)
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Background Information

 * $$C = \frac{Q}{V}$$ (Eq. 1.4.4)
 * $$I = C \frac{d v_C}{d t}$$ (Eq. 1.4.5)

Part A
Taking the derivative of (1.4.1)


 * $$V' = LC \frac{d^3 v_C}{d t^3} + RC \frac{d^2 v_C}{d t} + \frac{d v_C}{d t}$$ (Eq 1.4.6)

Rewrite equation (1.4.5)


 * $$I^{(n-1)} = C \frac{d^n v_C}{d t^n}$$ (Eq 1.4.7)

Substituting (1.4.7) into (1.4.6)


 * $$V' = L I^{(2)} + R I^{(1)} + \frac{1}{C} I^{(0)}$$

Rewrite


 * $$V' = L I'' + R I' + \frac{1}{C} I$$

Part B
Rewrite equation (1.4.4)


 * $$Q^{(n)} = C V^{(n)} \color{White} V' = L I'' + R I' + \frac{1}{C} I$$

Substituting (1.4.8) into (1.4.1)
 * $$V = L Q^{(2)} + R Q^{(1)} + \frac{1}{C} Q^{(0)}$$

Rewrite
 * $$V = L Q'' + R Q' + \frac{1}{C} Q$$