I'm stuck with a problem, it says:

Determine (if possible) a first order lineal partial differential equation that has as solutions

$\displaystyle u_{1}(x,y)=x^2+y+sin(x+2y)$

$\displaystyle u_{2}(x,y)=x^2+y+cosh(\sqrt[5]{(x+2y)})$

$\displaystyle u_{3}(x,y)=-(x+2y)^4+(x^2+y)$

I suppose that I am given the variable change $\displaystyle \xi(x,y) = x+2y$ so the pde should look like

$\displaystyle -2u_{x}(x,y) + u_{y}(x,y)+cu(x,y)=F(x,y)$

where $\displaystyle c \in \mathbb{R}$

Now, should I check the conditions $\displaystyle \{u_{i} \}$ impose on the pde and solve for $\displaystyle c$ and $\displaystyle F(x,y)$ ?

Thanks in advice