Page 304 - A Course in Linear Algebra with Applications

288 Chapter Eight: Eigenvectors and Eigenvalues

8. A certain species of bird nests in three locations A, B and
C. It is observed that each year half of the birds at A and half
of the birds at B move their nests to C, while the others stay
in the same nesting place. The birds nesting at C are evenly
split between A and B. Find the ultimate distribution of birds
among the three nesting sites, assuming that the total bird
population remains constant.
9. There are three political parties in a certain city, conserva-
tives, liberals and socialists. The probabilities that someone
who voted conservative last time will vote liberal or socialist
at the next election are .3 and .2 respectively. The proba-
bilities of a liberal voting conservative or socialist are .2 and
.1. Finally, the probabilities of a socialist voting conservative
or liberal are .1 and .2. What percentages of the electorate
will vote for the three parties in the long run, assuming that
everyone votes and the number of voters remains constant?

8.3 Applications to Systems of Linear Differential
Equations

In this section we show how the theory of eigenvalues
developed in 8.1 can be applied to solve systems of linear
differential equations. Since there is a close analogy between
linear recurrence relations and linear differential equations,
the reader will soon notice a similarity between the methods
used here and in 8.2.
For simplicity we consider initially a system of first or-
der linear {homogeneous) differential equations for functions
yi,..., y n of x. This has the general form

{ y'l = aiiVi + ••• + ainVn

= a nlyi + •• • +
y' n a nny n