The Single First Order Equation

Abstract

The first order equations, in general, present interesting geometric interpretations. It will be convenient then to restrict the discussion to the case of two independent variables, but it will be made clear that the theory can be extended immediately to any number of variables. We consider then equations of the form

$$ F(x,y,u,{u_x},{u_y}) = F(x,y,u,p,q) = 0 $$

((1))

where we have used the notation u_x = p, u_y = q. A solution z = u(x, y), when interpreted as a surface in three dimensional space, will be called an integral surface of the differential equation.