THE UNIVERSITY OF SYDNEY PURE MATHEMATICS Linear Mathematics 9 Tutorial 6 week 6 s Suppose that A and P are defined as follows: A and P Define a sequence of numbers { u n n } by u, u, u and for all n, a Check that u n+ u n+ + u n+ u n is the inverse of P by calculating its product with P b Evaluate P AP c Hence, find the eigenvalues of A and a basis of R which consists of eigenvectors for A d Express the recurrence relation for u n given above in matrix form e Find a formula for u n using parts c and d f Find lim u n n a Direct calculation shows that P P b Direct calculation shows that P AP { c By part b, the eigenvalues of A are, and, and,, of R consisting of eigenvectors for A un+ d Write u n u n+ Then u n u n+ un+ un+ u n+ u n+ u n+ u n+ u n+ + u n+ un u n+ un+ u n+ u u n n+ } is a basis That is, u n+ Au n e If {v, v, v } is a basis of R consisting of eigenvectors for A, with Av i λ i v i, then we know from lectures that the solution to the recurrence relation is of the form u n r λ n v + r λ n v + r λ n v, for some r, r, r R Hence, by part c, we have that u n r n + r n + r n Math 6: Tutorial 6 week 6 s SB and AM //9
Linear Mathematics Tutorial 6 week 6 s Page Now, u Hence, u u u r + r r r r P By definition, u n, so putting n we must have u n 8 + r n un+ u n+, so this means that u n f lim u n lim 8 + n n n n 8 n 8 + n n u n 8 n + n n r r r r P r r, and so At any time t, the populations of two species in symbiotic relationship each population supporting the other are denoted by xt and yt They are given by the system of linked differential equations: x t xt + yt y t xt + yt a Express the system in the form X t AXt, where A is a matrix and Xt xt yt b Find the eigenvalues and eigenvectors of A and write down the general solution of the system of linear differential equations c Find the particular solution of the system for the initial conditions x 6, y a We have, X X AX, where A b The characteristic equation of A is det λ λ λ λ λ 5λ 6 λ 6λ + So, the eigenvalues of A are λ and λ 6 If λ then we solve A + Iv : Hence, a basis for the eigenspace of A is { } If λ 6 then we solve A 6Iv : Math 6: Tutorial 6 week 6 s Page
Linear Mathematics Tutorial 6 week 6 s Page So, a basis for the 6-eigenspace of A is { } Therefore, the general solution of the differential equation X AX is X c e t + c e 6t, where c, c R c If t then x 6 and y Hence, 6 c + c Solving these two simultaneous equations gives c and c Consequently, X e t + e 6t Equivalently, xt e t + e 6t and yt e t + e 6t At any time t, the populations of two predator-prey species, xt and yt, are given by the following system of linked differential equations: x t xt yt y t xt + yt Suppose that, initially, x and y a Is xt the predator population or the prey population? b Find formulas for xt and yt at time t c After a long time, what is the proportion of predators to prey? a As yt appears with negative coefficient in x t, xt must be the population of the prey at time t and yt is the predator population b The system of differential equations can be rewritten as X AX, where X and A The characteristic equation of A is A λi λ λ λ 5λ + 6 λ λ, so the eigenvalues of A are λ and λ If λ then we find the corresponding eigenvectors by solving A Iv : Therefore, a basis for the -eigenspace is { } If λ then we solve A Iv : Therefore, the general solution of the differential equation X AX is X c e t + c e t, xt yt Math 6: Tutorial 6 week 6 s Page
Linear Mathematics Tutorial 6 week 6 s Page for constants c, c R When t we are told that X c + c Solving for c and c gives c c Hence, our particular solution is X e t + e t ; equivalently, xt e t + e t and yt e t + e t c If t is large then the term e t dominates e t, so xt e t and yt e t Therefore, in the long term the ration of predators to prey is approximately : Suppose that two tanks, each containing litres of salt water of a different concentration, are connected as shown below Fresh water liquid is pumped into tank A and then the water in the two tanks mixes as described by the diagram L/min 6 Tank L/min Litres A BL/min a Let xt and yt, respectively, be the number of grams of salt in tank A, tank B, respectively, at time t Find a system of differential equations in xt and yt b Find the general solution of the system of differential equations that you found in part a c Initially, tank A contains grams of salt while tank B contains grams of salt Determine the number of grams of salt in each tank at time t d Confirm that the concentration of salt in each tank approaches zero over a long period of time a The information given in the question says, mathematically, that dx 6 yt xt and dy 6 6 xt dt dt dx dt dy dt yt b Let Xt Then the equations in parta can be written as the single matrix equation X 6/ / AX, where A Hence, the general solution is 6/ 6/ X c e λ t v + c e λ t v, where v and v are eigenvectors of A with corresponding eigenvalues λ and λ We first fin dthe eigenvalues of A by solving the characteristic equation: 6 λ A λi 6 6 λ 6 + λ 6 λ + λ + 9 λ + λ + 8 Math 6: Tutorial 6 week 6 s Page
Linear Mathematics Tutorial 6 week 6 s Page 5 Therefore, the eigenvalues of A are λ and λ 8 When λ we find v by solving A + Iv : 8 6 8 Hence, a basis for this eigenspace is { }, so we take v When λ 8 we solve the equation A + 8Iv : 8 6 8 So, a basis for this eigenspace is { } and we may take v Consequently, the general solution is X c e t + c e 8t, for some constants c, c R c At t we have x and y That is, c + c Solving these simultaneous equations c 5 and c 5 Hence, our particular solution is X 5e t + 5e 8t That is, xt 5e t + 5e 8t and yt e t + 5e 8t d As t, the exponential terms approach zero so that X That is, xt and yt both approach over a long period of time 5 Suppose that two tanks, each containing litres of a mixture, are connected as shown in question Fresh water is pumped into tank A at a rate of 5 L/min Further, the mixture is pumped from tank A to tank B at a rate of L/min, and from tank B to tank A at 5 L/min Water then flows out of tank B at the rate of 5L/min Suppose that initially tank A contains 6 grams of salt while tank B contains pure water a Find a system of differential equations modelling the situation b Find the general solution of the system you found in part a c Determine the number of grams of salt in each tank at time t d Confirm that the concentration in each tank approaches zero in the long term a As in question, let xt and yt be the number of grams of salt in tanks A and B, respectively, at time t Then the system of differential equations for these tanks are the following: dx 5 yt xt and dy xt yt dt dt That is, X AX, where A 5 b We first find eigenvalues and eigenvectors of A: λ 5 λ λ + λ + λ + λ + Math 6: Tutorial 6 week 6 s Page 5
Linear Mathematics Tutorial 6 week 6 s Page 6 Hence, the eigenvalues of A are λ and λ If λ we find the corresponding eigenvectors by solving A + Iv : 5 So the eigenspace of A has basis { } If λ we solve A + Iv : 5 So the eigenspace of A has basis { } Therefore, the general solution of this system of differential equations is X c e 5t + c e 5t, for some c, c R c At time t we have X 6 That is, 6 c + c Solving these two equations we have c c Hence, our particular solution is X e 5t + e 5t That is,xt e 5t + e 5t and yt 6e 5t + 6e 5t d If t then xt and yt both approach zero because e 5t and e 5t Math 6: Tutorial 6 week 6 s Page 6