Question

Transform the given system into a single equation of second-order x'= -821 + 7:02 2 = -721 - 822- Then find 21 and 22 that also satisfy the initial conditions Then find 2, and are that alene 21 (0)=9 22 (0) = 2. Enter the exact answers. Enclose arguments of functions in parentheses. For example, sin (23).

Answer 1

aple - 80, +722 2.1.-11-802 ) -@) From a), 7 dil +801 =) a (a +r] =). Substitute an in equation (2) => [a, 427) = -73,- (0:) 09 -> [a," +8a1] =-70,- fo- 64 a, = -132,/ -> L+. - 1,- => 42," ta! +03, +2,50 = a." +862,1 + 1132, =0, This is second order equation.

Corresponding Ancillary equation is m² + 16m a 113 = 0 Then am – 16 + 141 - 8 + 7i ie m= -8+7i, -8-7i. Hence &, - est [c, cos (t) I ca sin (r)] Substitute &, in equation (3), then we get 2 = 4 [cest cas el) + ca e-st sinet] + & Enest coset) + Gest singt)] - [6, e-8 t - 7 sin (71) + C, cos(1) + - + ett + gest x 7 cos (21) + Q sin fhe-rett + & [cest cos (at) + Ca est sinet)]

-st sin(at) – ce et singt) usc, est cost) + GP-scos @t) - 24 + 4 ce3"cos@t) + cat sine t) - c, e8t sinft) + ca est cost) The initial conditions are 1,(0)=9 210)=2 = c est costat) & Ca est sin (7t) R, (o)=9 =) 9 - Ceo. cos (0) + Geo. sinco) = G, -) 959

= – ce etsin It) + ca est cosat) N 2 (0)=2 =) a = -c, e. sin (0) + ca eo cos(0) =) c2=2. e. ai = q est cos (At) + 2e-st sin ft) 2 = 26-st.cos(at) - gest sin (It)