DIFFERENTIAL EQUATIONS 
351 
ever, thus having the solution y — — 1, — co < x < -f-co; or we may 
leave the line at any point x x ^ x 0 , pass along the curve (3) to the 
other boundary, y — 1, and then continue forevermore on this line. 
Thus we see that through any point on the boundary of the strip 
pass infinitely many solutions of (i). 
Precisely similar results hold for (ii). The solution passing 
through a point (x 0 , y Q ) within the strip is given 
by 
(4) a; = cos" 1 y - cos- 1 y 0 + x 0 , 
where, as before, the principal value of each 
cos -1 is meant. And this solution continues Fig - 86 
along the boundary. 
The Solution of (1). We see now how to put together other solu 
tions of (1) than those given by (2). First, the solutions of (i) and 
(ii) just discussed are all 
solutions of (1). Secondly, 
we may start with the are 
AB of a solution of (¿), 
Fig. 87 proceed to the right of A an 
arbitrary distance, switch 
to a solution of (ii), follow the line y~ — 1, as far as we like, then 
switch to a solution of (i); and so on. 
Do we get, even in this way, all the solutions of (1) ? For an 
interior point (x 0 , y 0 ) any solution of (1) is given either by (3) or by 
(4) till it reaches the boundary. For a boundary point (x 0 , y 0 ) any 
solution, considered to the right of the point, either coincides with 
the boundary for an interval, or it has points distinct from the 
boundary in every neighborhood to the right of £ 0 . In the latter 
case, it must switch to a branch (3) or (4) to the right of x 0 , and the 
transition must obviously be made at the point x 0 . Similarly for 
the left-hand neighborhood of a boundary point. The solution of 
(1) is now complete. 
Example from Physics. Consider a simple pendulum, Introduc 
tion to the Calculus, p. 373. The differential equation of the first 
order * is 
(5) 2 (« 2 -0 2 ), n =yJ^, 
* This is the approximate equation for small arcs ; but the reasoning applies 
equally well to the accurate equation, 1. c. (2).