ABSTRACT
There are two distinct concepts of singularities in connection with differential equations: one relating to points and one relating to integrals. At a singular point of a differential equation, the highest order derivative either is not defined or takes multiple values; for example, it corresponds to a stagnation point of a flow. The singular integral of a differential equation is a particular solution that is not contained in the general integral; that is, it cannot be obtained by any choice of arbitrary constants. The consideration of special integrals is simplest for first-order differential equations and complements the general integrals. This chapter discusses certain types of first-order differential equations, namely Clairaut's and D'Alembert's forms, which have special as well as general integrals. The special integrals can occur not only for first-order differential equations, but also for higher-order differential equations.
