352
CALCULUS
and can be written down at once from the principle of Work and
Energy. Suppose that the pendulum gets stuck when it reaches its
highest point, and lodges there till some one releases it. This may
happen each time it comes to its highest point, and each time it may
remain at rest for an arbitrary interval of time. The equation of
Work and Energy is the same for this case as for the case ordinarily
considered, namely, (5).
From this it appears that (5) regarded as the mathematical for
mulation of the problem of Simple Pendulum Motion, is not ade
quate, since (5) admits other solutions, too. The same remark
applies to many of the deductions given in Mechanics, which are
based on the principle of energy and operate with differential equa
tions of the first order, which are not linear. On the other hand,
this situation cannot arise when the solution is based on Newton’s
Second Law of Motion and the formulation
-— = — y 6. (or = — sm 6.)
dt 2 l v l '
This differential equation has only one solution, and that, the solu
tion of the problem.
18. Continuation. The General Case. The central fact illustrated
by the example of § 17 may be stated as follows. The family of
solutions (2) have an envelope, namely, the lines y = 1 and y = — 1.
An arc of the envelope (a segment of either line) obviously must
also yield a solution of the differential equation; and yet this solu
tion is not contained in those given by (2). Such a solution is called
a singular solution.
We can generalize and say: Let a differential equation of the first
order be satisfied by a family of curves,
(1) y = <f>(x, c),
and let these curves all be tangent to a curve
(2) - y =
Through any point (x 0 , y 0 ) passes a curve, y = (f>(x,c 0 ), of the
family (1), and the function <f> (x, c 0 ) satisfies the differential equa
tion in the neighborhood o£ the point x = x 0 . If the curve (2) is not
contained in the family (1), it is called a singular solution.
For example, consider the differential equation
dy = 3yl
dx