Can someone help with these questions.

Question 1)

Consider the system _ = 11 + 30 , ddt 191 3'2 _2._-5 -+14 . dt" yl E2 11 30 _5 14) arez\\1 =4and A2 21 If we are told that the eigenvalues of the matrix A = ( and the associated eigenvectors are v1=(:>v2=(:>5 dy then two particular solutions to the system E = Ay are 2 u1(t>=e4t(l)and u2(t>=l a- 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 If\") = W1 ('5) + 19112 (t) for some oonstants o: and ,8 . If we are given the initial condition that y{0) = ( 35) then we can solve for o: and and get the solutions 311(1?) = |IE.'15'W=l 3120*) = | Recall: e. 1 - the Maple notation for the vector (2) is *exp{3*t) Consider the rst-order system of differential equations d'\"_1 = 351:1 (t) 8u2 (t) ddt % = 16011.1 (t) 37u2 (t). We can write this as the matrix equation _ A dt'_ u where A 2| l n E. The matrix A has eigenveotors v1 = ' i ' and v1 = (g) with corresponding eigenvalues A1 2| Number I and A2 2' Number I . Using all this information we can write out the solution 1 y = 4:]:(4)e)\\17t +u2(t) where u2(t) 2' In E. Recall: 1 2 - the Maple notation for the matrix (3 4) is {*exp (3*t). Consider the second-order ordinary differential equation 12 dy - y = 0 . dt dt 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 - cul + 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 21 (t) = 0 un (t ) e-t / 4 + B (3 ) et / 3 giving us y ( t ) = a Recall: the Maple notation for the exponential est is exp (2*t) .Consider the system (1'91 3 1 Ti =73\" + W (m 7E2\" W _3 l The associated matrix is A = 2 2 . Its eigenvalues, in decreasing order (Le. 1 1 A1 :> A2), are A1 2| Number | and A2 2' Number The corresponding eigenvectors are -n=l Insavm=l Inc 1 Recall: the Maple notation for the vector (2) is