t's
form. By differentiation in regard to x it gives

dp dp
p = p + x-- + f'(p)--,
dx dx

where

d
f'(p) = -- f(p);
dp

thus, either (i.) dp/dx = 0, that is, p is constant on the curve
satisfying the differential equation, which curve is thus any one of
the straight lines y = cx = f(c), where c is an arbitrary constant, or
else, (ii.) x + [f]'(p) = 0; if this latter hypothesis be taken, and p
be eliminated between x + f'(p) = 0 and y = px + f(p), a relation
connecting x and y, not containing an arbitrary constant, will be
found, which obviously represents the envelope of the straight lines y
= cx + f(c).

In general if a differential equation [phi](x, y, dy/dx) = 0 be
satisfied by any one of the curves F(x, y, c) = 0, where c is an
arbitrary constant, it is clear that the envelope of these curves,
when existent, must also satisfy the differential equation; for this
equation prescribes a relation connecting only the co-ordinates x, y
and the differential coefficient dy/dx, and these three quantities are
the same at any point of the envelope for the envelope and for the
particular curve of the family which there touches the envelope. The
relation expressing the equation of the envelope is called a
_singular_ solution of the differential equation, meaning an
_isolated_ solution, as not being one of a family of curves depending
upon an arbitrary parameter.

An extended form of Clairaut's equation expressed by

y = xF(p) + f(p)

may be similarly solved by first differentiating in regard to p, when
it reduces to a linear equation of which x is the dependent and p the
independent variable; from the integral of this linear equation, and
the original differential equation, the quantity p is then to be
eliminated.

Other types of solvable differential equations of the first order are
(1)

M dy/dx = N,

where M, N are homogeneous polynomials in x and y, of the same order;
by putting v = y/x and eliminating y, the equation becomes of the
first type considered above, in v and x. An equation (aB <> bA)

(ax + by + c)dy/dx = Ax + By + C

may be reduced to this rule by first putting x + h, y + k for x and y,
and determining h, k so that ah + bk + c = 0, Ah + Bk + C = 0.

(2) An equation in which y does not explicitly occur,

f(x, dy/dx) = 0,

may, th