Differential equation problem

MAT 275 Written Homework #9

7.1, 7.8 Due: November 25

Solve the following problems, showing any necessary work.

1. Let A =

[ 6 −9 1 0

] .

a. [1 point] The matrix A only has one eigenvalue. Find it and and an eigenvector of A.

b. [1 point] Find a generalized eigenvector of A.

c. [1 point] Solve the Initial Value Problem below.

~X′(t) =

[ 6 −9 1 0

] · ~X(t)

~X(0) =

[ 5 1

]

d. [1 point] Classify the phase portrait.

2. [1 point] Consider two interconnecting tanks. Tank 1 initially contains 750 L (liters) of water and 200 g of sugar, while Tank 2 initially contains 250 L of water and 100 g of sugar. Water containing 5 g/L of sugar is poured into Tank 1 at a rate of 40 L/m while the mixture flowing into Tank 2 contains a sugar concentration of 3 g/L and is flowing at a rate of 15 L/min. The mixture flows from Tank 1 to Tank 2 at a rate of 40 L/min. The mixture drains from Tank 2 at a rate of 50 L/min, of which some flows back into Tank 1 at a rate of 15 L/min, while the remainder leaves the tank. Let x and y, respectively, be the amount of sugar in each tank at time t. Set up BUT DO NOT SOLVE an initial value problem that models the flow process.