Question

I AM REALLY STRUGGLING ON THIS PROBLEM PLEASE HELP ME CORRECT AND NEAT WORK IS MUCH APPRECIATED THANKS

(1 point) a. Find the most general real-valued solution to the linear system of differential equations *' = [- xi(t) = C1 + C2 x2 (1) b. In the phase plane, this system is best described as a O source / unstable node O sink / stable node O saddle O center point / ellipses spiral source O spiral sink O none of these

Answer 1

a)
XE=; -G o 2 3 -5 Y- २ In-del=0 => -3-5-1 => 12+54+6=0 d=-2 -2=-3 d=-2 [A+QI]x = 0 2 2 -3 -3 J[3]= x=-4 v Fi]

d=-3 -O [A+31]x = 3 a -2 2 1-13) = -3 A= -24 3 let y=3 = -2 V = 3 2 X(E-) = Get [11]+sé - Pocit) [SCE), 13] = 4 -QE7 -at + -20 3t Зе b) bere eigenvalues acie real and d=-Qto and 2=-3 TO... Stable node.
b) here eigenvalues -2,-3 are reals and both are negative and distinct so in phase plane system is sink/stable node

Ans) sink/stable node

For part a we first find eigenvalues and eigenvectors of Matrix A.