I have a system of partial differential equations consisting of 6 equations on 9 variables.
$p = \frac{\partial f[x_1, x_2, x_3, y_1, y_2, y_3, z_1, z_2, z_3]}{\partial x_1};$
$q = \frac{\partial f[x_1, x_2, x_3, y_1, y_2, y_3, z_1, z_2, z_3]}{\partial x_2};$
$r = \frac{\partial f[x_1, x_2, x_3, y_1, y_2, y_3, z_1, z_2, z_3]}{\partial x_3};$
$o = \frac{\partial f[x_1, x_2, x_3, y_1, y_2, y_3, z_1, z_2, z_3]}{\partial y_1};$
$x = \frac{\partial f[x_1, x_2, x_3, y_1, y_2, y_3, z_1, z_2, z_3]}{\partial y_2};$
$a = \frac{\partial f[x_1, x_2, x_3, y_1, y_2, y_3, z_1, z_2, z_3]}{\partial y_3};$
$b = \frac{\partial f[x_1, x_2, x_3, y_1, y_2, y_3, z_1, z_2, z_3]}{\partial z_1};$
$c = \frac{\partial f[x_1, x_2, x_3, y_1, y_2, y_3, z_1, z_2, z_3]}{\partial z_2};$
$d = \frac{\partial f[x_1, x_2, x_3, y_1, y_2, y_3, z_1, z_2, z_3]}{\partial z_3};$
$equ1 = 2 x_1 p + 2 x_2 q + 2 x_3 r - y_1 o - y_2 x - y_3 a;$
$equ2 = 2 y_1 o + 2 y_2 x + 2 y_3 a -x_1 p - x_2 q - x_3 r - z_1 b - z_2 c - z_3 d ;$
$equ3 = 2 z_1 b + 2 z_2 c + 2 z_3 d -y_1 o - y_2 x - y_3 a ;$
$equ4 = (x_1 (x_1 + 2 x_2 + 2 x_3)) p + (x_2 (x_2 + 2 x_3)) q + (x_{3}^{2}) r - (y_1 (x_2 + x_3)) o - y_2 x_3 x;$
$equ5 = (y_1 (y_1 + 2 y_2 + 2 y_3)) o + (y_2 (y_2 + 2 y_3)) x + (y_{3}^{2}) a - (x_1 (y_1 + y_2 + y_3)) p - x_2 (y_2 + y_3) q - x_3 y_3 r - z_1(y_2 + y_3) b - z_2 y_3 c;$
$equ6 = (z_1 (z_1 + 2 z_2 + 2 z_3)) b + (z_2 (z_2 + 2 z_3)) c + (z_{3}^{2}) d - (y_1 (z_1 + z_2 + z_3)) o - y_2 (z_2 + z_3) x - y_3 z_3 a ;$
The first three equations are degree equations and show that the solution has degree zero. so somehow by changing the variables to $v_1 = \frac{x_2}{x_1}, v_2 = \frac{x_3}{x_1} , w_1 = \frac{y_2}{y_1}, w_2 = \frac{y_3}{y_1}, k_1 = \frac{z_2}{z_1} , k_2 = \frac{z_3}{z_1}$, we will have $3$ equations with $6$ variables.
I already did it by Mathematica and have the solutions for equations separately. But I need just one symbolic solution which satisfies in all the equations.
My proposition is that maybe MATLAB can do it? But I have no idea how to formulate this system on MATLAB. Can someone help me to formulate the coding on MATLAB please? Thanks!