Question

Solve the following initial value problem: 4-3 )+sin Compute Select one: A. B. C. D. (2e1

Answer 1

4) (-)-(-sin(t)) d-/3 -sin)- given x =Ax+f(t),here consider the characteristic polynomial det(A-A1)- f (t)- 26 3 4-3- therefore the eigen values are 5,-5 now we have to find the eigen vector for the corresponding eigen value 5 4 X1 X1 A-5I 4-3-5人x2 X2 therefore the equation form is xi 2x2 0,i 2x2 Here X2 is a free variable so, let X2 =S So, X2 now we have to find the eigen vector for the corresponding eigen value 4 X1 (4.sie--(3.4 "3+HE).。一(32)--8 İ 2)( )-. 1 1/2 8 4X1 therefore the equation form is x1+-X2-0 →X1 =-X2 Here x2 is a free variable so, let x2 -S 1/2 xi X2 1/2

therefore the general solution is 1/2 5t now the matrix X = (xi %) to findy, by using variation of parametres 5t 4 -sin (t) 0 2 and heref (t)- 2 so, now X-b- 4 2 0 es sin (t) 13 sin() e sin (t) dt 13 65

finally Xp = X1 X- Я--e": cos(t)+1 est sin (t) St ー-sin(t 13 13 65 general solution is x(r)eqre- -1/2) . | 26cos(t)-2-sin (r) 2H26 the general solution is x(t)-ces* -sin( t 13 By plugging the boundary value x(0) 26 3 (0).с.(2).eeG1/2)Jaco (아.헵(0 -sin (0 13 2 26 26 3 26 2 26 26 n comparing we get 13 26 cos(r)+26sin(t) therefore x(t)-26er(24e-stC1/2 - Sin (t 13 π)-26sin(r) 26 113 ーsin(r) 13 13 0 13 26