In general if a differential equation φ(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) + ƒ(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,