James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

588 Chapter 9 Differential Equations
(force equals mass times acceleration), we have
3
m d 2 x
dt 2
− 2kx
This is an example of what is called a second-order differential equation because it
involves second derivatives. Let’s see what we can guess about the form of the solution
directly from the equation. We can rewrite Equation 3 in the form
d 2 x
dt 2
− 2 k m x
which says that the second derivative of x is proportional to x but has the opposite sign.
We know two functions with this property, the sine and cosine functions. In fact, it turns
out that all solutions of Equation 3 can be written as combinations of certain sine and
cosine functions (see Exercise 4). This is not surprising; we expect the spring to oscillate
about its equilibrium position and so it is natural to think that trigonometric functions
are involved.
General Differential Equations
In general, a differential equation is an equation that contains an unknown function
and one or more of its derivatives. The order of a differential equation is the order of
the highest derivative that occurs in the equation. Thus Equations 1 and 2 are first-order
equations and Equation 3 is a second-order equation. In all three of those equations
the independent variable is called t and represents time, but in general the independent
variable doesn’t have to represent time. For example, when we consider the differential
equation
4
y9 − xy
it is understood that y is an unknown function of x.
A function f is called a solution of a differential equation if the equation is satisfied
when y − f sxd and its derivatives are substituted into the equation. Thus f is a solution
of Equation 4 if
f 9sxd − xf sxd
for all values of x in some interval.
When we are asked to solve a differential equation we are expected to find all possible
solutions of the equation. We have already solved some particularly simple differential
equations, namely, those of the form
y9 − f sxd
For instance, we know that the general solution of the differential equation
is given by
y9 − x 3
y − x 4
4 1 C
where C is an arbitrary constant.
But, in general, solving a differential equation is not an easy matter. There is no systematic
technique that enables us to solve all differential equations. In Section 9.2, how-
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