DIFFERENTIAL EQUATIONS 
353 
A family of solutions is seen to be 
y = (x- c) 3 . 
These curves are all tangent to the axis of x, and this line, y — 0, 
is seen to be a solution of the given differential equation. But it 
is not one of the above family ; it is a singular solution. 
Clairaut’s Equation. Consider the differential equation 
(3) 
where f(p) is continuous, together with f (p) and f"(p), and 
f"(p) =£ 0. 
The “ general solution ” (1) of (3) can be written down at sight: 
(4) 
y = cx+f(c) 
where c is an arbitrary constant. Thus we have a family of straight 
lines. 
This family, however, has an envelope; for, differentiate (4) par 
tially with respect to c, Chap. VIII, § 1: 
0 = x + /' (c). 
Thus the envelope is defined by the equations 
This curve represents a singular solution. 
2y 
IV. Solution by Series. Integrating Factor 
19. Bessel's Functions. Zonal Harmonics. The problems of 
Mathematical Physics lead to certain homogeneous linear differen 
tial equations of the second order with variable coefficients which 
are very simple functions. The most important equations of this 
class are: