Question

Question 6 Express the solution of the following system of equations as a linear combination of complimentary and particular solutions. **( )x+(3) Question 7: Find the particular solution for the following system of differential equations. 6 6t X + on (2010) -10t+4 Question 8: Use Variation of Parameters method to find particular solution for the following system X': 1-3 X + 30 2 -4

Answer 1

6 we have -1 2 x' 8 X + 3 - seep I Find the complementary solution (the system The characteristic equation 4 A-la? JA-AIT-0 2 =0 - 1 1-2 (-1-0 (1-2) +2=0 (+1) (1-1)+2=0 21+2=0 +1=0 d = i-i These are the eigenvalues of A find the rigerector of a for at i 1-1-i -1 P-e ]-[": :] R21 R2-211-8)ķ I -1 + RIT R 0 0 10 selwelline marix quation, we get -ltilin-50 N] [] 0

V +0-1+i) V2 =0 vi (1-4) v2 i V = 12 the Now, from just eigenrector solution of the system is. X(t) etit pit Vi 1- (ust disint) cost & int -i cost tisint cost + isint cost + sint sint Wy(t) E cost tti cost sint Thes, the complementary solution in given by r(t) = G, + c2 Sint-cost er cost+ sint cost sunt

Stopli het were partituras solution 460)å [:] pizto] to Beyou plugging into the system dut's simplify the notation, 1 2 1X + 3 - X! Ax + Now, -get lets plug the values To = Aă + - ģ -> Aa = 2 བད az - 1 1-1 2 -2 - a2 -3 Consledu 2 8 1 N 1993 0 1973 19 weget, 424 422 x(t)= 14/3 19/

There you, the genual solution of this system u alt) = xect) + xp (t) cost t sint ta a(t) ost Isint - cost sint 1473 119/3