Can someone help with these questions.

Question 1)

Consider the system _ = 11 +30 , dd: yl 3'2 _ 5 + 14 . dt '91 1'12 11 30 If we are told that the eigenvalues of the matrix A = ( 5 14 )areA1 =4andA221 and the associated eigenvectors are 2 3 v1: 1 andv2= 1, then two particular solutions to the system % = Ay are 2 u1{t>=e4*(1)and u2{t>=l @- Note: we are looking for a particular solution corresponding to the second eigenvalue {i.e. your solution should not contain arbitrary constants). The general solution to the system has the form 3'05) = D$11105} + 3112(3) for some constants c: and . If we are given the initial condition that y(0) = ( 35) then we can solve for a: and ,8 and get the solutions 910$} = l Iand 3126} = l Recall: a. 1 - the Maple notation for the vector (2) is *exp {3*t) Consider the rst-order system of differential equations (3%? : 3511.1 (t) 8u2 (t) {2+3 2 1601.91 (t) 37u2 (t). We can write this as the matrix equation @_ citAu where A 2| E] E. g 1 J 1 The matrix A has eigenvectors v1 = 4 and v1 = (5) with corresponding eigenvalues A1 2| Number | and A2 2' Number | . Using all this information we can write out the solution 3': a (i) ellt +u2(t) @- where u2(t) 2' Recall: 1 2 - the Maple notation for the matrix (3 4) is {*exp (3*t). Consider the second-order ordinary differential equation 12 day dy dt dt - y= 0 . If we set uj (t) = y(t) and u2 (t) = dy dt then we get the system of first-order equations du1 -au] + buz dt duz = cu + duz dt where a = Number b Number , C: Number and d = Number Using the same technique as in the last question we can construct the general solution uj (t ) ( 242 ( 2 ) 14 + B ( ? ) @ / 3 giving us y ( t ) = a Recall: the Maple notation for the exponential e- is exp (2*t) .Consider the system @1 3 1 Ti $in + it? 6392 TF2\" W _3 l The associated matrix is A = 2 2 . Its eigenvalues, in decreasing order (i.e. 1 1 A1 22> A2], are A1 2' Number ' and A2 2| Number The corresponding eigenvectors are vla lnEmmu=l lus- 1 Recall: the Maple notation for the vector (2 ) is 00 is' 926} E