Question

a. Find the most general real-valued solution to the linear system of differential equations a' 2 -9 -2 2. 21(t) 음을 + C2 22(t) b. In the phase plane, this system is best described as a O source / unstable node sink / stable node saddle center point / ellipses spiral source spiral sink none of these preview ang

Answer 1

solution: -9 given B's ( 7 ) -2 ř=A for general solution ; a INI-AI X+2 At2 (1 +233 +9 = 0 (172)3= -9 A+ 2 = ± 3 5-2 + 3 i 21= -2+31) Az - -2-3 Ala .-2+3 i (I-A)X =0 3 [8 3:11 13:43 ع) ه → zix, +90% -K, +31 og x = Zil - DC hence eigen rector [:]

I) for A= -2-3i [ AI-A][x] = -3 9 SC 13] -z -zix, +90₂=0 -3,-3ixe so 00=-Bil = hence eigen rector [3] Thus the general solution: -3 X C, 2+31) [3] 62** = 8-2-30) į + zi $2+3) -31 12-31) X= X,17). C t ₂ Dit -2-31) 2+31)

-2 - 9 X 1) $73 -2 So we need do some points in the phase plane, system, We'll plug them into the right side of the do this for a couple of points. x=1 ñ -3 77 - 13 → 18 +(371:1-6 - [ + ][3] -11 [1 :][13] -(124) L rx 3=11 - Doing this for a large number of points in the phase plane;

4.5 4 3.5 3 2.5 2 1.5 ✓ 1 0.5 0 1 -0.5 -1 1 1 → → 1 -1.5 → → → → -2.5 > > 1 -3.5 -4 -4.5 n -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1.5 2 2.5 3 3.5 4 4.5

स्स्स्स्स 0.75 ہہہہہہہہ یہ ہے کہ اس کا هااااااااااا یہ ہے کہ اس ک ک ک ک کک ک ک ک ک ک | 05 کعبے سے و 0.25 -0.25 ححححححججددد ,جدید حججدددددد حج ويحدد -0.5 부부 -0.75 뺙 075 un 025 0 -0.25 -0.5 -0.75

45 4 35 3 25 2 15 ft: 05 0 / -05 1 -1.5 2 -25 - -3.5 45 LT 4.5 -- -3.5 - -2.5 2 -15 | -0.5 0 0.5 1 15 2 2.5 3 35 45