Example 5

When a spring of natural length L is compressed a distance x it exerts a force F = -kx. The negative sign indicates that the force is in the opposite direction from x (Figure 8.6.3).

Figure 8.6.3

When x is negative the spring is expanded and the equation F = -kx still holds.

Suppose a mass m is attached to the end of the spring and at time t = 0 is at position x₀ and has velocity v₀. The motion of the mass follows the differential equation

The general solution is

x = a cos cot + b sin ωt where Using the initial conditions, the motion of the mass is

This function is periodic with period 2π/ω, so as expected the mass oscillates back and forth.

In the following second order equation, hyperbolic sines and cosines arise. The general solution of the differential equation

d²y/dx² = y

is

y = a cosh x + b sinh x.

We see that cosh x and sinh x are solutions because

, ,

, .

Another solution is e^x. Note that