Question

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Assume that a 2 x 2 matrix A has eigenpairs X ati, a -1, B 5, Problem 8V Problem 9 Problem 10... Problem 11 Problem 12 V-atbi, a = uz = Let the function x be the solution of the differential system x = Ax given by x(t) = (velit + v. e). Write this solution in terms of real valued fundamental solutions, draw its phase portrait, and then answer the questions below (a) Find the rotation direction of the solution values x() as increases. Clockwise (b) Find the limit as t o o of the solution values (0)| (c) Determine the vector lim None (d) Find the limit as t -00 of the solution values (2) Zero (e) Determine the vector lim * st) None (1) Characterize the trivial solution Xo Source

Answer 1

Solenen A be a 2x2 matrix has eigen vialetto paine + = + (3 , 4 = -1, 2 = 5 W = atbi: Q=1376-14] 4 = 102.4 let X be a solution of the differential equation X'= AX given by XIH) = (viellett vedt System X141 = + (0) = (2) ****131 - 14:3).*** *1# 0 [4749 6:5+) +[3]le : X122823 x 14) = of 607541 - 2 sim 5 tt ) Bin 5)] x 12) = o-t ra los 58-iy sin 5t L 66 54 tisin 5t) t=0 then first Gordinate is (2 0) da I then first Coordinate is (0, -12) So, it's looks in to votate in clockwise direction. (a)"

11241 = Ji-? (2 cost is first)*+ (e*(695+ihast Je* (C's+ + (4+1) finºst 120.89)1 = pott ese cast - (b) So, as fy too 1130:9|| yo Pr) cm 12654 - i fins + pia) 12210)] = lume 51 15 0754 CS5H i Shin 5t) py fou 4 4 - $2 U, ti 42 (d) : 1414 10 = pot + 6954 4 4 as to 7-00 i t -0 en you 711241) ya

re) = al (B) = host Cm line 14 - Niet bescher / (6) 65+ Pilihing gel (8) +i (3) So, we get i Y(-1) = 24, -illa