Question

4y (1 point) Find the solution to the linear system of differential equations 6. 3. satisfying the initial conditions (0) = -9 and y(0) -7 (t) - u(t)

Answer 1

volution page - xco) = -9 yco) - 7 - 62-49 32-4 with Let x= x ¿t) y (t) can be written as given dynamical system x = Х .+ Xcol 7 3 step. I Now bet A= 6 -4 3 - 6-X 3 -4 - (6-) (-1-^2 - (-12) -- (A²-sxto) (x-3) (A-2) =) = 2,3 are eigen rates of A Step II re going to find eigenvectors of x =24 o for x=2 consider the system we are 6-2 O 4 24 O Perform: R₂ 22 - 3 R • ] [ h is free Variable

& we get page x - 20 Let X2 = t (od is free variable) for some ter. 42 t 't f t 24 eigenveeter of A comesponding oo to eigenvalue xai x=3 consider the system now for -4 -1-3 3 O 3 -4 4 perform, RR R ķR, X / 7 2413 24 O o 1. 2 x is free variable het xet for some tGR f x - 4x = 0 2 X= 46 t ra + [] ir eigenester of a correspon sponding to pa 3

3 page ystem is ezt XCt) = ta e * so general solution to the given dynomical est pas where ci fa are arbitary constant Now X (o)=-9 - 7 4/3 + -9 7 G+ 4 = =-9 С + С, = -7 G (4-1) = -2 T2 = -6 put it in 2 f put GEN 4 =-6 in - X(t) = - pet[:] [415] = xot) = - et gest 0 2t t6031 - e - 8 e e+ Es solution of = fact) = {yct) = - eet 6e3t given dynamical Sitem