Applications of L inear Equations with Constant Coefficients 2976.2 Higher Order Differential Equations
In this section we will present some applications which require the solution
of linear differential equations with constant coefficients of order greater than
two. And we will show how to solve a linear system of differential equations
with constant coefficients by writing the system as a single higher order linear
differential equation with constant coefficients.
A Coupled Spring-Mass System Suppose a mass m1 is attached to
one end of a spring with spring constant k 1. The other end of this spring is
attached to a fixed support. A second mass m2 is attached to one end of a
second spring with spring constant k2. The other end of the second spring is
attached to the bottom of mass m 1 and the resulting system is permitted to
come to rest in the equilibrium position as shown in Figure 6.7.
System at
EquilibriumBoth Springs in
Stretched PositionFigure 6. 7 A Coupled Spring-Mass SystemFor i = 1 , 2 let Yi represent the vertical displacement of mass mi from its
equilibrium position. As before we will assign downward displacement from
equilibrium to be positive and upward displacement from equilibr ium to be
negative. Applying Newton's second law of motion and assuming no damping
is present, it can be shown that t he equations of motion for this coupled