Question 1 1 pts What is the system of ODEs a; = 3x1 - x2 + cos(t), a2 = 21 + 4x2 + t - e t in matrix form X' = AX + G? ox - (2 7 ) x + (cos( ) Ox' = (3 1 x + 05 () " = (1 37) x + ((0(2. ) O X' cos ( t ) it - et 1 ) x + ( 2 ) Question 2 1 pts Suppose that X1, X2, and X3 solve the IVP's X' = AX1 , X1(to) = U1 X'2 = AX2, X2(to) = U2 X? = AX3 , X3(to) = U3 . Find a solution to the IVP X' = AX, X(to) = c1 U1 + c2 U2 + c3 U3 . O X = c1X1 + c2X2 + C3X3 O X = qj U 1 + c2 U2 + (3 UsQuestion 3 1 pts Consider the system X' = AX, where A is a n x n matrix with constant entries. If I give you n + 1 solutions to the system - say X1, X2, - - - 5Xn+1 - can the X,- be linearly independent? Answer true if they can be linearly independent. False otherwise. True Q False Question 4 1 pts Consider the system X' = AX, where A is a n x n matrix with constant entries. If I give you 71. solutions to the system - say X1, X2, ~ - - , X\" - and | tell you that these X,- span all solutions to the system. Are they linearly independent? Answer true if they are. False otherwise. 0 True False Question 5 1 pts Consider the system X' = AX, where A is a n x n matrix with constant entries. If I give you n i 1 solutions to the system - say X1, X2, - - - 5Xn+1 - and | tell you that these X,- are linearly independent, can these X,- form a basis of all solutions to the system? Answer true if they can, false otherwise. 0 True False Question 6 1 pts Let A be a constant entry 3 x 3 matrix and consider the system X' = AX. Suppose X1, X2, and X3 are solutions to 1 3 1 this system with X1(2.5) = 71 ,X2 (2.5) = 1 and X3 (2.5) = 73 .Are these solutions linearly 2 0 4 independent? Answer true if they are, false otherwise. 0 True 0 False Question 7 1 pts 3 6 What is the general solution to the system X' : ( ) X? 5 4 0 KM : (-56) +2th (1) 0 X0) : ((1)) +28% ((1)) 0 X6) =clet (56) +626; (1) O X(t):616' (3) met (2) Question 8 1 pts What is the general solution to the system X' = X ? x(t) = cient (1 )+ catext Ox(t ) = ce2t (?) O X (t ) = cie2t (1) + cze" (2) O No solutions Question 9 1 pts What is the (real) general solution to the system X' = 4 17 X O sin(4t) sin(4t) X(t) = cle3t - + cos( 4t) 17 sin(4t) + Cze 3t cos(4t) sin (4t) 4 Ox (t) = ciest ( 8 ) + cze3t ( " ) O X (t) = cle(3+4i)t + cze(3-4i)t No solutions