Explanation Formula used: The differential equation for the electric circuit is given by L Q ″ + R Q ′ + 1 C Q = E ( t ) with the initial conditions Q ( t 0 ) = Q 0 , Q ′ ( t 0 ) = I 0 where R, C, L and Q are resistance, capacitance, inductance and charge, respectively. If the roots of the characteristic equations are real and distinct, then the solution is given by y = A e r 1 t + B e r 2 t where r 1 and r 2 are distinct roots. Calculation: Given that a series circuit has a capacitor of 10 − 5 F , a resistor of 3 × 10 2 Ω and an inductor of 0.2 H . That is, C = 10 − 5 F , R = 3 × 10 2 Ω and L = 0.2 H . From the above formula, the differential equation is given by L Q ″ + R Q ′ + 1 C Q = E ( t ) . Assume that the initial charge on the capacitor is 10 − 6 C and there is no initial current. That is, E ( t ) = 0 and Q 0 = 10 − 6 C . Substitute these values in the differential equation L Q ″ + R Q ′ + 1 C Q = E ( t ) . 0.2 Q ″ + ( 3 × 10 2 ) Q ′ + 1 10 − 5 Q = 0 0.2 Q ″ + 300 Q ′ + 10 5 Q = 0 2 Q ″ + 3000 Q ′ + 100000 Q = 0 Q ″ + 1500 Q ′ + 500000 Q = 0 Thus, the simplified differential equation for the given capacitor is Q ″ + 1500 Q ′ + 500000 Q = 0 with initial conditions Q ( 0 ) = 10 − 6 H and Q ′ ( 0 ) = 0 . The corresponding characteristic equation is r 2 + 1500 r + 500000 = 0 . Obtain the roots of the above equation as follows. r 2 + 1500 r + 500000 = 0 r 2 + 1000 r + 500 r + 500000 = 0 r ( r + 1000 ) + 500 ( r + 1000 ) = 0 ( r + 500 ) ( r + 1000 ) = 0 r = − 500 , − 1000 Since the roots of the characteristic equation are real and different, the solution of the differential equation is given by Q ( t ) = A e − 500 t + B e − 1000 t

10th Edition

ISBN: 9780470458327

Author: William E. Boyce, Richard C. DiPrima

Publisher: Wiley, John & Sons, Incorporated

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Chapter 3.7, Problem 12P

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The expression for charge Q on the capacitor at any time t.

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Chapter 3.7, Problem 11P

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Chapter 3 Solutions

Elementary Differential Equations

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The expression for charge Q on the capacitor at any time t .

The expression for charge Q on the capacitor at any time t.

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