On a system of PDE

Question

I would like to know what is the set of solutions to the following PDE. I think it consists of just constants, but I need help to prove. Let $f_1(p_1,p_2)$ and $f_2(p_1,p_2)$ be two functions. The system of equations on $\mathbb{R}^2$ is just:

$$p_1 \partial_{p_1} f_1 + p_2 \partial_{p_1} f_2=0 $$

$$ p_1 \partial_{p_2} f_1 + p_2 \partial_{p_2} f_2 =0 $$

So nobody complain, I want the solutions to be at least of class $C^2$.

Answer 1

$$p_1 \partial_{p_1,p2} f_1 + p_2\partial_{p_1,p2} f_2= -\partial_{p_1} f_2 \tag{$\partial_{p_2}\,eq1$}$$

$$ p_1 \partial_{p1,p_2} f_1 + p_2 \partial_{p1,p_2} f_2 = -\partial_{p_2} f_1 \tag{$\partial_{p_1}\,eq2$} $$

So $$ \partial_{p_1} f_2 = \partial_{p_2} f_1 \tag{*}$$ Can you take from here? Now you have PDEs in one unknown. Solve and check whether their solution satisfies (*)...