EXAMPLE 4 Consider the electric circuit shown in the figure and modeled by the differential equation below. di dt + RI =

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Example 4 Consider The Electric Circuit Shown In The Figure And Modeled By The Differential Equation Below Di Dt Ri 1 (34.5 KiB) Viewed 23 times

EXAMPLE 4 Consider the electric circuit shown in the figure and modeled by the differential equation below. di dt + RI = E(t) Find an expression for the current in a circuit where the resistance is 12 , the inductance is 4 H, a battery gives a constant voltage of 36 V, and the switch is turned on when t = 0. What is the limiting value of the current? SOLUTION With L = 4, R = 12, 4 di + 121 36 or dI dt and the initial value problem is dI dt = We recognize this equation as being separable, and we solve it as follows: fat dt (9 - 31 # 0) --In 19 31 = t + C 19 - 31 = dI 9 - 31 I(t) = = 9 - 31 = ± I = 3 - Since I(0) = 0, we have 3- and E(t) = 36, the equation becomes = 9 - 31 I(0) = 0. The limiting current, in amperes, is lim I(t) = lim (3 - 3e-3t) = t → co t → co = Ae-3t Ae-3t = 0, so A = = 3 - 3 lim e-3t = t → co and the solution is