Reduction of 2nd order PDE to a first order equations system

[Auteng]

I want to convert this linear second order general form PDE to two equations:
$ϕ_{xx}+bϕ_{xy}+cϕ_{yy}+dϕ_x+eϕ_y+fϕ=g(x,y)$

Converted equations:

$a_1 u_x+b_1 u_y+c_1 v_x+d_1 v_y=f_1$

$a_2 u_x+b_2 u_y+c_2 v_x+d_2 v_y=f_2$

I want to find parametric values of $a_1 ...f_2$

How can I do it?

Hint:

$ϕ_{xx}+ϕ_{yy}=0$

$u=ϕ_x , v=ϕ_y$

$ϕ_{xx}=u_x,ϕ_{yy}=ϕ_y$

$u_x+v_y=0$

$u_y-v_x=0$

 

[andrewkirk]

1. The converted equations don't look right, as they do not incorporate the function $g$.

2. What is the justification for the first line of the Hint: $\phi_{xx}+\phi_{yy}=0$? It is not derivable from the given equation.

 

[Auteng]

No that hint is a typical example