Question

Solve the system.

(1 pt) Solve the system with x(0) = Give your solution in real form. Xi = 1. Describe the trajectory An ellipse with clockwise orientation

Answer 1

(1) aneous The given system of simultaneous dineal differential equations is og et = -9 310 - fl) with Sle) = ) - Let a= ail, therefole flow equation (1), We have T-907347 terms, we have J-300, +902 equating the coffesponding = -99, +302 and t = -30014902 > ali+90-30 =0 and 300, to! -902=0

(2) - Da,+901, 302 = 0 and Bout. Dx-942 = 0 where I = → (D+90-3042 = 0 – 13) 30 dit(0-9).Xazo 14 opacting equation (3) by (D-9) and equation (4) by 3 we have (8-9) (D+9).04,- 3(0-9)%=0 -75 9ox + 3(0-9)1.20 - 16) adding equations (5) and (6), we have (3-9) (D+994,+904, = 0 3 (0-81001+ gol = 0 = ($%81+9018, = 0 = (2+9) 41=0 (7) equation (7) is a lineat differential equation with constant Coefficients. . Antilliary equation of (7) is given by mat9 = 0

Wyp = -9 = 972 C od (3) m= 737 8:22_203) =m= 0f3i = xtiß whole x=0,B=3 in complimentry function (CFO) = eat ( 9 CopBt+G-DixBt) = et G, Co33t+Q Sin3t) = e la Col3t+G sinist) u. Cof. = C, Cosst to singt also the right hand side of equation (7) | is zelo, therefore the particular integral, P. I = 0 e Complete solution of equation (7) is given by 1 = Corot P. Io a = C, Cost+ & Sin3t - (8) Where Ci and Co are arbitraly Constants.

differentiating equation (8), with despect to t, we have a= -3C, Sin3t +36 6633t — (9) Now feloly equation (3), we have 3i - (+1) = 1 = 34= (-36, Sin3t +36 6083€) +91 6, 6033t + G Sin3t) {flon equations (3) and (9)} = 196, +3G) Co834 +1+34+9 9J3435 394 = 3(136+6) Col3+ +/-C;+36) 3434] do= 136+G) C033+ +(+36) Sin3t Now from equation (%), we have lloj Falco) – 4-17 - Ull(o)=-| 210) = -2 (11)

putting t=0 in equations (8) and (10) we have Hilo) = 6; Co.8(0) + G SH (O) = -1 from (103 and qulo) =f3c,+G) Casco) + (-C,+36] Sinoy = -2 {flom (12)} (1) +G(0) --| and (30+ C) (1) + (-0,+3G) (0) = -2 S CHO) = 1, Simlolz & cito =-11 and (30+ 2) to = -2 and 130+ 6) = -2 re 3-1) + 6 =-2 {2C, =-1} te -3+6=-2 le C=3-2=1 We have C = -1, CQ = 1 putting C, = -1 and 2 = 1 in equations (8) and 110), we have

(6) Yo = - Colgt + Singt and 2 = (-3+1) Cosat+ (143) sinst 2 s= – Cosatt sinst and a = -2 Cosat + 4 sinst hence & lego - Co33+ + 21:3+ ] 1-2 Cast + 4 siust is a solution of the simultaneous lineal equations de given system of differential