Page 307 - A Course in Linear Algebra with Applications

8.3: Applications to Systems of Linear Differential Equations 291

To find the original functions yi, simply use the equation Y =
SU to get
n n

3=1 3=1

Since we know how to find S, this procedure provides
an effective method of solving systems of first order linear
differential equations in the case where the coefficient matrix
is diagonalizable.

Example 8.3.1
Consider a long tube divided into four regions along which
heat can flow. The regions on the extreme left and right are
kept at 0°C, while the walls of the tube are insulated. It is
assumed that the temperature is uniform within each region.
Let y(t) and z(t) be the temperatures of the regions A and
B at time t. It is known that the rate at which each region
cools equals the sum of the temperature differences with the
surrounding media. Find a system of linear differential equa-
tions for y[t) and z(t) and solve it.

\
/ A B
0° "s n°
m° z(t)°

According to the law of cooling

y' =(z-y) + {0-y)

Z' =(y- Z) + (p- Z)