Special type II: PDEs not involving the independent variables explicitly

In this case, a given PDE is of form

$$f(u, u_x, u_y) = 0.$$ (1.29)

The auxiliary equation in Charpit's method

$$\frac{dx}{f_{u_x}} = \frac{dy}{f_{u_y}} = \frac{du}{u_x f_{u_x} + u_y f_{u_y}} = \frac{du_x}{-u_x f_u} = \frac{du_y}{-u_y f_u}$$

From the last relation

$$u_x = a u_y ~\hspace{1cm} (a~ ).$$ (1.30)

Solve () and () for $u_x$ and $u_y$, one get the required complete integral.

Example 1.6.3 (Already solved)   Find the complete integral of $u u_{x}^{2} + u_{y}^{2} = 4$.

Answer $(1 + a^2 u) = \pm 3a^{3} x \pm 3 a^2 y + 3a^2b)^{\frac{2}{3}}$

Example 1.6.4   Find the complete integral of $u^2 u_{x}^{2} + u_{y}^{2} = 1$.

Answer $(1 + a^2 u^2) - \log{(au + \sqrt{1 + a^2u^2} )} = 2a(ax+y +b).$

Answer $au\sqrt{1 + a^2 u^2} + \log{(au + \sqrt{1 + a^2u^2} )} = \pm 2a(ax+y +b).$