EXAMPLE 4 Consider the electric circuit shown in the figure and modeled by the differential equation...

 

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EXAMPLE 4 Consider the electric circuit shown in the figure and modeled by the differential equation below. LI+RI = E(t) dt Find an expression for the current in a circult where the resistance is 12 01, the inductance is 4 H, a battery gives a constant voltage of 48 V, and the switch is turned on when t-0. What is the limiting value of the current? SOLUTION With L-4, R-12, and E(t)- 48, the equation becomes +127-48 or di dt and the initial value problem is di 12-31 I(0) - 0. We recognize this equation as being separable, and we solve it as follows: J12 -In112-311 <-+ C di 12-31 112-371- I dt 12-31 - t e <-12-31 I(t)-4-4e 1-4-8 -3c Since 7(0) = 0, we have 4- B-e The limiting current, in amperes, is (12-310) x X Ae-t x -Ae-t lim_1(t) = lim (4-40-³). -4-4 lim # (15 t→∞ -0, so A-4 -3t4 X and the solution is EXAMPLE 4 Consider the electric circuit shown in the figure and modeled by the differential equation below. LI+RI = E(t) dt Find an expression for the current in a circult where the resistance is 12 01, the inductance is 4 H, a battery gives a constant voltage of 48 V, and the switch is turned on when t-0. What is the limiting value of the current? SOLUTION With L-4, R-12, and E(t)- 48, the equation becomes +127-48 or di dt and the initial value problem is di 12-31 I(0) - 0. We recognize this equation as being separable, and we solve it as follows: J12 -In112-311 <-+ C di 12-31 112-371- I dt 12-31 - t e <-12-31 I(t)-4-4e 1-4-8 -3c Since 7(0) = 0, we have 4- B-e The limiting current, in amperes, is (12-310) x X Ae-t x -Ae-t lim_1(t) = lim (4-40-³). -4-4 lim # (15 t→∞ -0, so A-4 -3t4 X and the solution is

Answer rating: 100% (QA)

SOLUTION Solution L with 15 RI Et L4 4 dl 12 48 d R12 de or de dt and ... View the full answer