Question

let F(u,v) where u=u(x,y,z) and v=v(x,y,z), further let z=z(x,y). show that on elimination of the arbitrary function F we obtain the first order partial differential equation;

Pp+Qq=R

Answer 1

Clearly $F(u, v) = 0$ is an implicit function.

$$0=F_x=\dfrac{\partial F }{\partial u}\dfrac{\partial u}{\partial x}+\dfrac{\partial F }{\partial v}\dfrac{\partial v}{\partial x}$$

$$0=F_y=\dfrac{\partial F }{\partial u}\dfrac{\partial u}{\partial y}+\dfrac{\partial F }{\partial v}\dfrac{\partial v}{\partial y}$$

$$\dfrac{\partial F }{\partial v}=-\dfrac{\dfrac{\partial F }{\partial u}\dfrac{\partial u}{\partial y}}{\dfrac{\partial v}{\partial y}}$$

Substitute

$$\dfrac{\partial F }{\partial u}\dfrac{\partial u}{\partial x}-\dfrac{\dfrac{\partial F }{\partial u}\dfrac{\partial u}{\partial y}}{\dfrac{\partial v}{\partial y}}\dfrac{\partial v}{\partial x}=0$$

$$\dfrac{\partial u}{\partial x}\dfrac{\partial v}{\partial y}-\dfrac{\partial u}{\partial y}\dfrac{\partial v}{\partial x}=0$$

$$\dfrac{\partial u}{\partial x}=\dfrac{\partial u}{\partial x}+\dfrac{\partial u}{\partial z}\dfrac{\partial z}{\partial x}$$

$$\dfrac{\partial v}{\partial x}=\dfrac{\partial v}{\partial x}+\dfrac{\partial v}{\partial z}\dfrac{\partial z}{\partial x}$$

$$\dfrac{\partial u}{\partial y}=\dfrac{\partial u}{\partial y}+\dfrac{\partial u}{\partial z}\dfrac{\partial z}{\partial y}$$

$$\dfrac{\partial v}{\partial y}=\dfrac{\partial v}{\partial y}+\dfrac{\partial v}{\partial z}\dfrac{\partial z}{\partial y}$$

Then

$$(\dfrac{\partial u}{\partial x}+\dfrac{\partial u}{\partial z}\dfrac{\partial z}{\partial x})(\dfrac{\partial v}{\partial y}+\dfrac{\partial v}{\partial z}\dfrac{\partial z}{\partial y})$$

$$-(\dfrac{\partial u}{\partial y}+\dfrac{\partial u}{\partial z}\dfrac{\partial z}{\partial y})(\dfrac{\partial v}{\partial x}+\dfrac{\partial v}{\partial z}\dfrac{\partial z}{\partial x})=0$$

$$\dfrac{\partial u}{\partial x}\dfrac{\partial v}{\partial y}+\dfrac{\partial u}{\partial z}\dfrac{\partial v}{\partial y}\dfrac{\partial z}{\partial x}+\dfrac{\partial u}{\partial x}\dfrac{\partial v}{\partial z}\dfrac{\partial z}{\partial y}+\dfrac{\partial u}{\partial z}\dfrac{\partial v}{\partial z}\dfrac{\partial z}{\partial x}\dfrac{\partial z}{\partial y}$$

$$-\dfrac{\partial u}{\partial y}\dfrac{\partial v}{\partial x}-\dfrac{\partial u}{\partial z}\dfrac{\partial v}{\partial x}\dfrac{\partial z}{\partial y}-\dfrac{\partial u}{\partial y}\dfrac{\partial v}{\partial z}\dfrac{\partial z}{\partial x}-\dfrac{\partial u}{\partial z}\dfrac{\partial v}{\partial z}\dfrac{\partial z}{\partial x}\dfrac{\partial z}{\partial y}=0$$

$$(\dfrac{\partial u}{\partial y}\dfrac{\partial v}{\partial z}-\dfrac{\partial u}{\partial z}\dfrac{\partial v}{\partial y})\dfrac{\partial z}{\partial x}+(\dfrac{\partial u}{\partial z}\dfrac{\partial v}{\partial x}-\dfrac{\partial u}{\partial x}\dfrac{\partial v}{\partial z})\dfrac{\partial z}{\partial y}$$

$$=\dfrac{\partial u}{\partial x}\dfrac{\partial v}{\partial y}-\dfrac{\partial u}{\partial y}\dfrac{\partial v}{\partial x}$$

$$P=\dfrac{\partial u}{\partial y}\dfrac{\partial v}{\partial z}-\dfrac{\partial u}{\partial z}\dfrac{\partial v}{\partial y}$$

$$Q=\dfrac{\partial u}{\partial z}\dfrac{\partial v}{\partial x}-\dfrac{\partial u}{\partial x}\dfrac{\partial v}{\partial z}$$

$$R=\dfrac{\partial u}{\partial x}\dfrac{\partial v}{\partial y}-\dfrac{\partial u}{\partial y}\dfrac{\partial v}{\partial x}$$

$$p=\dfrac{\partial z}{\partial x}, q=\dfrac{\partial z}{\partial y}$$

$$Pp+Qq=R$$