On singular solutions of Clairaut equations in the theory of ordinary differential and partial derivative equations

Background. There is no problem in finding a general solution to the ordinary differential Clairaut equation. The corresponding procedure is described in details in the theory of ordinary differential equations. Except of general solution being the family of linear functions, special (singular) solutions for the ordinary differential Clairaut equation may exist for which are no general methods to find them. This is evidenced by a very meager list in the accessible scientific literature of the types of Clairaut equations for which special solutions can be explicitly constructed. Therefore, it seems as an actual task finding and studies special solutions to the Clairaut equations. Goal of the present paper is finding and studies of special solutions to the Clairaut equations in the theory of ordinary differential equations and of partial differential equations and setting relations among special solutions to the Clairaut equation in the theory of ordinary differential equations and of partial differential equations.
Results. We discuss basic concepts and solutions to the Clairaut equations in the theory of ordinary differential equations and of partial differential equations. We find relations between singular solutions to the Clairaut equations in the theory of ordinary differential equations and of partial differential equations.
Conclusions. It is proved that there exists the relation between singular solutions to the Clairaut equations in the theory of ordinary differential equations and of partial differential equations in the class of special dependences of right hand side of the Clairaut in the theory of partial differential equations. The number of such relations is defined by the number of known special solutions to the Clairaut equations in the theory of ordinary differential equations and all their possible combinations.