In the  rst chapter of this book, we derived the following equation based on Newton’s

second law to compute the velocity y of a falling parachutist as a function of time t

[recall Eq. (1.9)]:


dt 5 g 2


m y (PT7.1)

where g is the gravitational constant, m is the mass, and c is a drag coef cient. Such

equations, which are composed of an unknown function and its derivatives, are called

differential equations. Equation (PT7.1) is sometimes referred to as a rate equation

because it expresses the rate of change of a variable as a function of variables and pa-

rameters. Such equations play a fundamental role in engineering because many physical

phenomena are best formulated mathematically in terms of their rate of change.

In Eq. (PT7.1), the quantity being differentiated, y, is called the dependent variable.

The quantity with respect to which y is differentiated, t, is called the independent vari-

able. When the function involves one independent variable, the equation is called an

ordinary differential equation (or ODE). This is in contrast to a partial differential equa-

tion (or PDE) that involves two or more independent variables.

Differential equations are also classi ed as to their order. For example, Eq. (PT7.1)

is called a ! rst-order equation because the highest derivative is a  rst derivative. A

second-order equation would include a second derivative. For example, the equation

describing the position x of a mass-spring system with damping is the second-order


m d 2x

dt 2 1 c


dt 1 kx 5 0 


the gravitational constant
Order Now on