Question

the solution need to be in terms of sin and cos

(1 point) Consider the linear system -3 -2 = r. 5 3 Find the eigenvalues and eigenvectors for the coefficient matrix. 1 11 = -I V1 = help (numbers) help (matrices) -(3-1)/2 and 1 12 = -. help (numbers) help (matrices) 1 V2 = -(3+i)/2 Find the real-valued solution to the initial value problem x1 = -321 – 2x2, x = 5x1 + 3x2, x1(0) = 2, x2(0) = -5. Use t as the independent variable in your answers. xi(t) = 2*cos(t) + 5* sin(t) help (formulas) x2(t) - -5* cos(t) + 2* sin(t) help (formulas)

Answer 1

x IC A 5 3 x=-30-20, x(0)=2 a = 5+ 3X (0) = -5 > for eigen values, (A-111=0 and o de ce, do 5 3-d for eigen vectors, A-df)x=0 1 3+4) (3-1) do, the solution corresponds to these eigen values and eigen vectors, -it 3 (t) = çe - (372) it + 3G 0) = 2, 3, (0) = -5 & gello) 3+x Gta = 2 5 - (31) -CE) - (3-2) +(3+i) = 10

multiplying equi ② with ~(3ti) and then adding ③ we get, G-i) 6 (3+i) G = 10-2(372) - - 2i6= 2 [5-3 -2] G = è (Qri) = 1 + 21 from ean. O a = 2-6=2-(1 + 2) = 1-21 -it ix(+) = (1+98) e - + (1-28) dit (3 te 24 (I) = (1 + 2) (costi sint) + (1-21) (cost & isint) Da ( #) (costri sint + Ricost + 2 8int) et (cost t e sint - eicost + esint) () = 2 cost + 4 sint and oxy(t) = (1+2:) ( 33) (cosf - isine) + 3) (cost-isine) + (1-21) (**) cost tisint) 5 cost - 5 sint X (H)