Question

Find the most general real-valued solution to the linear system of differential equations x⃗ ′=[1−34−6]x⃗ .x→′=[14−3−6]x→.

⎡⎣⎢⎢[ x1(t)x1(t) ⎤⎦⎥⎥] x2(t)x2(t)

=c1=c1

⎡⎣⎢⎢[ ⎤⎦⎥⎥]

+ c2+ c2

⎡⎣⎢⎢[ ⎤⎦⎥⎥]

a. Find the most general real-valued solution to the linear system of differential equations a = [_3_-4). 1 4 3 - 6 xit) = C1 + C2 22(t) b. In the phase plane, this system is best described as a source / unstable node sink / stable node saddle center point / ellipses spiral source spiral sink none of these

Answer 1

O X' ne rigenvalues of the mateix A [26] [3:1-74 ]=0 A-I = 0 CK 4 oldule 1-3 -6-1 - (1-1) (6 + d) + 12=0 FI+A) (6+1) +12=0 -6-176d+ 1² + 12=0 12 +51 +6=0 - 1²+24+31+ 6 = 1(1+2 +3 (2+2)=0 and (=-3 Eigen vectors cosresponding to these eljanvallues- A) for d = -2 AX = dx 1=-2 u - 2 지 U2 02 - 34 = 4x2 2 2 = 3 x1 mit 4 2 =-2017 emparing

-3% -6% = -2x2 - 3x1 = 4x₂ 21 4 So, let 4 = 4 10 8₂ = -3. 80, Vi 3 4 21 -3 -3-6 X2 22 رند (B) for d=-3. AX= xx (-3 nit 4H2= - 3x By comparing both sides - 34 - 642 = - 322 x + 422 = - 341 = 4x, = - 4X2 14 =-X2 ( )= (-1) General solution is written as timeau combination of elgen solutions Salt) 10, Jean 201 -(1)**4/4)* Uz (it)

since her both li and d₂ all less than fixed point. So, we can say fined point is the stable zero (dira <O). This means the function is to, at to all the trajectories approaches decay function. suponentially the point. Trajectories approaches the origin tangent to slowes eigen direction (one with smaller I'll) Hence phase plane is best described as sink / stable node.