# Solving functional equations with aditional constraints

i am trying to solve a complicated system of differential equations, which involves several hundred of equations. Luckily they can be all solved seperately, but there are to many to do it by hand. The first of them are of the form

$P1*A[x_1,x_2,x_3,x_4,x_5]+P2*B[x_1,x_2,x_3,x_4,x_5]+P3*C[x_1,x_2,x_3,x_4,x_5]+P4*D[x_1,x_2,x_3,x_4,x_5]+P5*E[x_1,x_2,x_3,x_4,x_5]=P6$

subject to

$B[x_1,x_2,x_3,x_4,x_5]=A[x_2,x_1,x_3,x_4,x_5]$,

$C[x_1,x_2,x_3,x_4,x_5]=A[x_3,x_2,x_1,x_4,x_5]$

$D[x_1,x_2,x_3,x_4,x_5]=A[x_4,x_2,x_3,x_1,x_5]$

$E[x_1,x_2,x_3,x_4,x_5]=A[x_5,x_2,x_3,x_4,x_1]$

where all the Pi are given Polynomials in the $x_i$. If this is not mathematical exact enough, the system is given by the equations 3.14-3.16 in https://arxiv.org/pdf/1606.07376.pdf The problem now is how to implement these constraints. Taking

$5 x1^4 A[x1, x2] + 5 x2^4 B[x1, x2] == x1^6 x2^6$

as a toy example,which is by hand easy to solve by $A=(x1^2*x2^6)/10$, I tried to insert the conditions directly in the equation, i.e.

DSolveValue[5 x1^4 A[x1, x2] + 5 x2^4 A[x2, x1] == x1^6 x2^6, A[x1, x2], {x1, x2}]


which does not work as it says the variables are not ordered correctly. Then I tried to implement it as a condition,

DSolveValue[
{5 x1^4 A[x1, x2] + 5 x2^4 B[x1, x2] == x1^6 x2^6,A[x1, x2] == B[x2, x1]},
A[x1, x2],
{x1,x2}]

DSolveValue[
5 x1^4 A[x1, x2] + 5 x2^4 B[x1, x2] == x1^6 x2^6, A[x1, x2],
{x1, x2},
DirichletCondition[A[x1, x2] == B[x2, x1], true]]


which also does not work as in the first case, It simply does nothing and in the second one says it cannot use the $x_i$ as variables. Is there a way to solve such problems in Mathematica?

• I don't see anything that looks a PDE in this post. – m_goldberg Oct 12 '17 at 15:03
• I am searching for functions of several variables, which for me was a PDE. After looking up the exact definition i see that the examples i gave did not contain derivatives and therefore are probably not PDEs, but i do not know a better name for them. Nonetheless, in the later iterations of the system also derivatives of the A appear and then there are PDEs to solve with these conditions. But the issue still remains, how can I tell Mathematica to solve such systems. – Rohbar Oct 12 '17 at 16:01
• I believe, MMA cannot do this (cannot into functional equations). See here. But I'd like to see this resolved. – LLlAMnYP Oct 13 '17 at 8:22