Question

Please solve ASAP, I only need the correct answer.

(3 points) Find a particular solution to the system of equations x = -72x - 320y – 5e-8, y = 16x + 72y. Both of your functions must be correct to receive credit. x(t) = ((8/5)-5)^(-8t)-4e-(8t)-25t*e^(-8t) y(t) = e^(-8t)+e (8t)+5te^(-8t)

Answer 1

- 727- 3207 584 We have given the non- homogeneous dyaline of eqreations x yl 16x + 727 the given system may be written in the matrix notation as X(t)= AXL+ F(!) (t) å(t) Now where A(72 2 72 -320 16 72 ýt) F(L) (-58®t)X(t) = (yo)

at Let X(t) = Ké be a solution of the homogeneous eqreation $(t) = AX(t) connesponding to the given non- homogeneous equaloon. K = CK). Then act) = kjent, y(t) = kg eat.

OP 9 or, OP Now the characteristic eqreation of A is 1-72-2 -320 = 0, 7 16 72-2 2² - Trac (A)2 + det (A) = 0 22 - (-72+72)2 + (-5184 +5120) 22-64 = 0, OY, 7 = 18. Now the eigen vector connesponding to 1 = 8 is given by A X = 8 X *-72 -320 8x y 8 y -723-3204 Org -44 16x + 72Y 13-47 Let us take y =6, x = -4e. OY, (-12 12 org 87, 89, on,

an eigen value eigen vector connesponding to the 7 = 8 is (4) Again the eigen vector comesponding to the 2 = -8 is value eigen given by, AX = -8 x ong -72 -320 16 72 9)(3) *(-9%) Ong -72x - 3207 16% +72y = -82, on, 2=-59 -89 or, X=-54 Let us take y=d, then 23-5d. to the an eigen vector connesponding 2=-8 is Now the fundamental matrix of the homogeneous equalion Ř CE) = A X(t) is M(+) -5 eigen value -4 get -5084 est e 8t det (MCL)) - 4 +5 = 1 # O, for all te[a, b], [M(+)]-1 exist. where [a, b] is a closed interval.

1 [MC4)]=1 adj {M(t)} det & M(t)} i l'est est 5e87 8t -4 e [MCL)] PPL 8t -4081 284 5287 est -408t FCL) C5 5 ***") fet [Mocas]Srcus du So ( 5 100) There 2 du. to 16th du There we take to =1 5 166 16 5t Thus solution f(t) of the given homogeneous equeation is geven by MCE) St [MCU)]-' FCU) du a mom- *() to

orn Plt) 8t -40 -5787 5 16 @ 166_5 8t è st 5t 8t 87 +20 e 2 st - 25 të - 5 est st - 5e - t + 5e 16 105 -st st + 20e 11 8t 16 85-87 est - 54 the required solution ( Particular) of the given system is 105 287 XCD YCE) st + 20 e st 5e 85 ē8t ans). 16