Differential Equations, 3rd Edition

5Applications of the Second-Order Theory

DOI: 10.1201/9781003214526-5

• Vibrations and oscillations

• Electrical current

• Newton's law of gravitation

• Kepler's laws

• Higher-order equations

5.1 Vibrations and Oscillations

When a physical system in stable equilibrium is disturbed, then it is subject to forces that tend to restore the equilibrium. The result can lead to oscillations or vibrations. It is described by an ordinary differential equation of the form

$$\begin{array}{l}\frac{d^{2}x}{d{t}^2}+p(t)·\frac{dx}{dt}+q(t)x=r(t).\hfill \end{array}$$

In this section we shall learn how and why such an equation models the physical system we have described, and we shall see how its solution sheds light on the physics of the situation.

5.1.1 Undamped Simple Harmonic Motion

Our basic example will be a cart of mass ...