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Title: Diagonalization and differential equations
Description: Linear algebra course
Description: Linear algebra course
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Diagonalization and Differential
Equations
6
...
The ideas are not used elsewhere in this book
...
At a
initial time, t
=
0 (in hours), the concentration of salt in tank Y is different from the
concentration in tank Z
...
The two tanks are joined by pipes; through one
pipe, solution is pumped from Y to Z at a rate of 20 L/h; through the other, solution is
pumped from Z to Y at the same rate
...
in each tank at time
y(t)
(z/1000)
(20)(z/1000)
dy -0
...
02z
...
(20)(y/1000)
t, z(t)
t (y/1000)
and let
Then the concentration in Y at time
be the
is
kg/L
...
Then for tank Y, salt is fl
...
Since the rate of change is measured by the derivative, we have
By consideration of Z, we get a second differential equation, so
+
=
are the solutions of the system of linear ordinary differential equations:
and
dy -0
...
02z
dtdz
z
...
02y
-0
dt
d [y] [-0
...
02] [y]
dt z 0
...
02 z
+
=
=
...
...
'
t
1s
convernent
to
rewnte
t
1s
system
m the form
h
I
=
·
How can we solve this system? Well, it might be easier if we could change vari
2
2
[-0
...
0022 -0
...
02]
...
04,
[11 -11]
ables so that the
A
=
x
matrix is diagonalized
...
ith p
'W
=
=
l
2
[
4
...
...
Introduce new coordinates
tion
...
0
by the change of coordinates equation, as in Sec-
[�]
d [y*] - [y*]
dt z* z*
Substitute this for
-P
on both sides of the system to obtain
AP
Since the entries in Pare constants, it is easy to check that
d
dt
[yz**] - dtd [yz**]
-P-
-P
Multiply both sides of the system of equations (on the left) by p-i
...
04z*
dt
dt
-
=
and
-
=
These equations are "decoupled," and we can easily solve each of them by using simple
one-variable calculus
...
So, from
b is a constant
...
The only
kx are exponentials of the form
-0
...
To determine the constants
the amounts y(O) and z(O) at the initial time t
all t
...
04t
+ be-0
...
This is the general
and b, we would need to know
0
...
A Practical Solution Procedure
The usual solution procedure takes advantage of the understanding obtained from this
diagonalization argument, but it takes a major shortcut
...
ce,i1
[:]
...
We find the two eigenvalues A
...
2 and the corresponding
eigenvectors v1 and v2, as above
Title: Diagonalization and differential equations
Description: Linear algebra course
Description: Linear algebra course