Question

Chapter 7, Section 7.5, Question 32a An electric circuit is described by the system of differential equations (0) - -1 CR (). lu Find the general solution of this equation if R1 - 1 ohm, R2 = 39 ohm, L = 6 henrys, and 6 35 farad.

Answer 1

solution The given electric Circuit is described by the system of equations differential Ri L +0: 00 CR2 Given Ri= 1 ohm, R2 = 32 ohm, L=6 hennys and C= -farad 6 35 Now, Ź = 33 faried I = - hennys R = 1 ohm ohm/hennys 37 ohm-farad = CR2 6 35 35 37 Now the above system of equation becomes -16 dt -- 35 6 35 37 6

d! Ć I tv dt du dt 35 1 - 3 tv => -- - d = 331 - 27 NOW from (1) 6 dl =-1-v dt V =-1- odl dt Now putting the value of 'v' in the equation -(ii) we have dl *(-4-44)-(1-44 dl di dt 6 d²1 dtz 351 +371 +372 - 38 dhe d21 dt2 - 121=0

6 d²1 +38 di + 121=0 dtz dt (iii) m being Let l=emt (whene Constant) be a solution of the homogenious equation of (ii) . Then its auxiliary equation Coresponding lis 6m² + 38m + 12 = 0 - 38 I J (38)2_4x6*12 => m= 2 x 6 m - 381 51444-288 12 - m -38 1 J 1156 12 - 381 34 - m 12 m -38-434 12 - 4 12 - 3 and -38-34 12 -72 =-6 m = 12 m = -6, - - -

- 6t + C₂ e Thus the Genen al solution of -(iii) is of the form l(t) = Cie (iv) where C., C2 are arbitrary constant Now differentiating (iv) w.nitt we have =-q ett ocze -6t di dt Now (i) fr rom dl dt 6 6 6 dl =-1-V > dt V =-1- 6 di dt ſt -6t С, е = -Cie => V - 1²t -6t -6c2e - 1²t - 6t С, е - - - - facie +36 Cest => V =-cie -6t +35cge - v=c, e ist

the + C2 est Now dl dt dv dt - 1²t General solution of the system of differential equation + 1 - V is 351 - 3 tu ce 0 - ht Cie 43502 e 6t st - 6t =c е e & C2 35 The option Ć is correct answer.