Special type IV: Clairaut equations

Definition 5 (Clairaut type equations)   A first order PDE is said to be of Clairaut type, if it can be expressed in the form

$$u = x u_x + y u_y + f(u_x, u_y).$$

The corresponding auxiliary eqs. in Charpit's method:

$$\frac{dx}{x + f_{u_x}} = \frac{dy}{y + f_{u_y}} = \frac{du}{x u_x + y u_y + u_x f_{u_x} + u_y f_{u_y}} = \frac{du_x}{0} = \frac{du_y}{0}$$

From the last two relations

$$u_x = a, \hspace{1cm} u_y = b, \hspace{1cm} ( a, b~$$ are constants$$).$$

Put these values in the given PDE, one get required complete integral

$$u = ax + by + f(a, b).$$

Example 1.6.6   Find the complete integral of

$$(u_x + u_y)(u - x u_x - y u_y) = 1.$$

Solution Rewriting is as,

$$u = x u_x + y u _y + \frac{1}{u_x + u_y}.$$

This is in Clairaut's form. Therefore, the required complete integral is

$$u = ax + by + \frac{1}{a+b}.$$