Page 324 - Advanced engineering mathematics

304 CHAPTER 10 Systems of Linear Differential Equations

We can also write the fundamental matrix
t 6t
e e
(t) = t 6t
−3e /2 e
in terms of which the general solution is X(t) = (t)C.
If we write out the components individually, the general solution is
t
x 1 (t) = c 1 e + c 2 e 6t
3
x 2 (t) =− c 1 e + c 2 e .
6t
t
2
EXAMPLE 10.7
Consider the system
⎛ ⎞
5 14 4
X = 12 −11 12 ⎠ X.

⎝
4 −4 5
The eigenvalues of A are −3,1,1. Even though there is a repeated eigenvalue, in this example,
A has three linearly independent eigenvectors. They are
⎛ ⎞
1
associated with eigenvalue − 3
⎝ 3 ⎠
1
and
⎛ ⎞ ⎛ ⎞
1 −1
and associated with eigenvalue 1.
⎝ 1 ⎠ ⎝ 0 ⎠
0 1
The general solution is

⎛ ⎞ ⎛ ⎞ ⎛ ⎞
1 1 −1
t
e −3t t e .
X(t) = c 1 ⎝ 3 ⎠ + c 2 ⎝ 1 ⎠ e + c 3 ⎝ 0 ⎠
1 0 1
We also can write the general solution X(t) = (t)C, where
⎛ −3t t t ⎞
e e −e
(t) = 3e −3t e t 0 ⎠ .
⎝
e −3t 0 e t
EXAMPLE 10.8 A Mixing Problem
Two tanks are connected by pipes as in Figure 10.1. Tank 1 initially contains 20 liters of water
in which 150 grams of chlorine are dissolved. Tank 2 initially contains 50 grams of chlorine
dissolved in 10 liters of water. Beginning at time t = 0, pure water is pumped into tank 1 at a rate
of 3 liters per minute, while chlorine/water solutions are exchanged between the tanks and also
flow out of both tanks at the rates shown. We want to determine the amount of chlorine in each
tank at time t.
Let x j (t) be the number of grams of chlorine in tank j at time t. Reading from
Figure 10.1,

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October 14, 2010 20:32 THM/NEIL Page-304 27410_10_ch10_p295-342