Special type I: First order PDEs involving only $u_x$ and $u_y$

Under special circumstances, elaborate Charpit's method gets simplified, and one can solve such PDE easily. For this type of PDEs

$$f(u_x, u_y) = 0$$ (1.27)

The auxiliary equations in Charpit's method reduce to

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

From $du_x = 0$, one have $u_x = a$ (constant). The corresponding value of $u_y$ can be obtained from () as

$$f(a, u_y) = 0 ~ ~ u_y = Q(a) = ,$$ (1.28)

and the required solution is

$$u = ax + Q(a)y + b, \hspace{1cm} (b~$$ is constant$$).$$

Note Here one have taken $u_x=0$ to get the compatible equation (). In some examples, it is advantageous to start with $u_y=0$.

Example 1.6.2   Find the complete integral of the equation $u_x u_y = 1$.

Solution In this case

$$u_{x} = a,$$

$$u_{y} = \frac{1}{a} = Q(a).$$

$\therefore$ The complete integral is

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