Answer to Question #252278 in Differential Equations for pde

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

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

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

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

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

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

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

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

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

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

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

$Pp+Qq=R$