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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##

##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##

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