Page 312 - A Course in Linear Algebra with Applications

296 Chapter Eight: Eigenvectors and Eigenvalues

Finally, suppose that initial conditions j/i (0) = 1 and
V2 (0) = 0 are given. We can find the correct values of c\ and
by substituting t = 0 in the expressions for y\ and j/2, to get
c 2
2x
ci = 1 and C2 = — 1. The required solution is y\ = (1 — x)e
2x
and ?/2 = — x e .
The next application is one of a military nature.
Example 8.3.4
Two armored divisions A and B engage in combat. At time t
their respective numbers of tanks are a(t) and b(t). The rate
at which tanks in a division are destroyed is proportional to
the number of intact enemy tanks at that instant. Initially
A and B have ao and bo tanks where ao > &o- Predict the
outcome of the battle.
According to the information given, the functions a and
b satisfy the linear system
a' = -kb
b' = -ka

where k is some positive constant. Here the coefficient matrix
is
0 -k'
A
~ ' -k 0
The characteristic equation is x 2 — k 2 = 0, so the eigenvalues
are k and —k and A is diagonalizable. It turns out that

where S = ( . If we set F = , the system of

differential equations becomes Y' = AY. On writing U =
X
S- Y, we get U' = DU. This is the system
u' = ku
v' = —kv