Page 339 - Advanced engineering mathematics

10.5 Applications and Illustrations of Techniques 319

10.5 Applications and Illustrations of Techniques

This section presents some examples involving mechanical systems and electrical circuits, whose
analysis gives rise to systems of differential equations. We have previously applied the Laplace
transform to solve such systems. Here we will apply matrix methods.

EXAMPLE 10.16 A Mass/Spring System
We will analyze the system of three springs and two weights shown in Figure 10.2, which dis-
plays the spring constants and the mass of each weight. At time 0, the upper weight is pulled
down one unit and the lower one is raised one unit, then both are released. We want to know the
position of each weight relative to its equilibrium position at any later time.
The initial value problem to be solved is

y =−8y 1 + 2y 2 ,
1

y = 2y 1 − 5y 2 ,
2

y 1 (0) = 1, y 2 (0) =−1, y (0) = y (0) = 0.
1
2
Begin by converting this system of two second-order differential equations to a system of four
first-order differential equations by putting
x 1 = y 1 ,
x 2 = y 2 ,

x 3 = y ,
1
and
x 4 = y .

2

k = 6
1

y 1
k = 2
2

m = 1
y 2 2
= 3
k 3

FIGURE 10.2 Mass/spring
system of Example 10.16.

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