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Question: How build up a system of equations of the coefficient ?

Question:How build up a system of equations of the coefficient ?

salim-barzani 1555 Maple 2024
parameters substitution pde differential-equations duplicate-question
April 26 2025
3

I try to construct a system of coefficient but  i don't know why distribute of them is not working, beside this there is any other way for build this kind of systems 

> 

restart
> 

with(PDEtools)
> 

with(LinearAlgebra)
> 

NULL
> 

with(SolveTools)
> 

_local(gamma)
> 

``

`There is no more prime differentiation variable; all derivatives will be displayed as indexed functions`

 (1)
> 

declare(u(x, y, z, t))

u(x, y, z, t)*`will now be displayed as`*u

 (2)
> 

declare(f(x, y, z, t))

f(x, y, z, t)*`will now be displayed as`*f

 (3)
> 

pde := 9*(diff(u(x, y, z, t), t, x))+diff(u(x, y, z, t), `$`(x, 6))-5*(diff(u(x, y, z, t), `$`(x, 3), y)+diff(u(x, y, z, t), `$`(y, 2)))+15*((diff(u(x, y, z, t), `$`(x, 2)))*(diff(u(x, y, z, t), `$`(x, 3)))+(diff(u(x, y, z, t), x))*(diff(u(x, y, z, t), `$`(x, 4)))-(diff(u(x, y, z, t), x))*(diff(u(x, y, z, t), x, y))-(diff(u(x, y, z, t), `$`(x, 2)))*(diff(u(x, y, z, t), y)))+45*(diff(u(x, y, z, t), x))^2*(diff(u(x, y, z, t), `$`(x, 2)))+alpha*(diff(u(x, y, z, t), `$`(x, 2)))+beta*(diff(u(x, y, z, t), x, y))+delta*(diff(u(x, y, z, t), x, z))

9*(diff(diff(u(x, y, z, t), t), x))+diff(diff(diff(diff(diff(diff(u(x, y, z, t), x), x), x), x), x), x)-5*(diff(diff(diff(diff(u(x, y, z, t), x), x), x), y))-5*(diff(diff(u(x, y, z, t), y), y))+15*(diff(diff(u(x, y, z, t), x), x))*(diff(diff(diff(u(x, y, z, t), x), x), x))+15*(diff(u(x, y, z, t), x))*(diff(diff(diff(diff(u(x, y, z, t), x), x), x), x))-15*(diff(u(x, y, z, t), x))*(diff(diff(u(x, y, z, t), x), y))-15*(diff(diff(u(x, y, z, t), x), x))*(diff(u(x, y, z, t), y))+45*(diff(u(x, y, z, t), x))^2*(diff(diff(u(x, y, z, t), x), x))+alpha*(diff(diff(u(x, y, z, t), x), x))+beta*(diff(diff(u(x, y, z, t), x), y))+delta*(diff(diff(u(x, y, z, t), x), z))

 (4)
> 

``
> 

oppde := [op(expand(pde))]; u_occurrences := map(proc (i) options operator, arrow; numelems(select(has, [op([op(i)])], u)) end proc, oppde); linear_op_indices := ListTools:-SearchAll(1, u_occurrences); pde_linear := add(oppde[[linear_op_indices]]); pde_nonlinear := expand(simplify(expand(pde)-pde_linear))

9*(diff(diff(u(x, y, z, t), t), x))+diff(diff(diff(diff(diff(diff(u(x, y, z, t), x), x), x), x), x), x)-5*(diff(diff(diff(diff(u(x, y, z, t), x), x), x), y))-5*(diff(diff(u(x, y, z, t), y), y))+alpha*(diff(diff(u(x, y, z, t), x), x))+beta*(diff(diff(u(x, y, z, t), x), y))+delta*(diff(diff(u(x, y, z, t), x), z))

  

15*(diff(diff(u(x, y, z, t), x), x))*(diff(diff(diff(u(x, y, z, t), x), x), x))+15*(diff(u(x, y, z, t), x))*(diff(diff(diff(diff(u(x, y, z, t), x), x), x), x))-15*(diff(u(x, y, z, t), x))*(diff(diff(u(x, y, z, t), x), y))-15*(diff(diff(u(x, y, z, t), x), x))*(diff(u(x, y, z, t), y))+45*(diff(u(x, y, z, t), x))^2*(diff(diff(u(x, y, z, t), x), x))

 (5)
> 

H := u(x, y, z, t) = 2*(diff(ln(f(x, y, z, t)), x))

u(x, y, z, t) = 2*(diff(f(x, y, z, t), x))/f(x, y, z, t)

 (6)
> 

H1 := int(pde_linear, x)

(diff(u(x, y, z, t), z))*delta+alpha*(diff(u(x, y, z, t), x))+beta*(diff(u(x, y, z, t), y))-5*(int(diff(diff(u(x, y, z, t), y), y), x))+9*(diff(u(x, y, z, t), t))+diff(diff(diff(diff(diff(u(x, y, z, t), x), x), x), x), x)-5*(diff(diff(diff(u(x, y, z, t), x), x), y))

 (7)
> 

L := eval(H1, H) = 0

-18*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), t))/f(x, y, z, t)^2-12*(diff(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x), x))*(diff(f(x, y, z, t), x))/f(x, y, z, t)^2-30*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x))*(diff(diff(f(x, y, z, t), x), x))/f(x, y, z, t)^2-20*(diff(diff(diff(f(x, y, z, t), x), x), x))^2/f(x, y, z, t)^2+60*(diff(diff(f(x, y, z, t), x), x))^3/f(x, y, z, t)^3-240*(diff(f(x, y, z, t), x))^6/f(x, y, z, t)^6+10*(diff(diff(diff(f(x, y, z, t), x), x), x))*(diff(f(x, y, z, t), y))/f(x, y, z, t)^2+30*(diff(diff(diff(f(x, y, z, t), x), x), y))*(diff(f(x, y, z, t), x))/f(x, y, z, t)^2+30*(diff(diff(f(x, y, z, t), x), x))*(diff(diff(f(x, y, z, t), x), y))/f(x, y, z, t)^2-60*(diff(f(x, y, z, t), x))^2*(diff(diff(f(x, y, z, t), x), y))/f(x, y, z, t)^3+60*(diff(f(x, y, z, t), x))^3*(diff(f(x, y, z, t), y))/f(x, y, z, t)^4+60*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x))*(diff(f(x, y, z, t), x))^2/f(x, y, z, t)^3-240*(diff(diff(diff(f(x, y, z, t), x), x), x))*(diff(f(x, y, z, t), x))^3/f(x, y, z, t)^4+240*(diff(diff(diff(f(x, y, z, t), x), x), x))*(diff(f(x, y, z, t), x))*(diff(diff(f(x, y, z, t), x), x))/f(x, y, z, t)^3+720*(diff(diff(f(x, y, z, t), x), x))*(diff(f(x, y, z, t), x))^4/f(x, y, z, t)^5-540*(diff(diff(f(x, y, z, t), x), x))^2*(diff(f(x, y, z, t), x))^2/f(x, y, z, t)^4-60*(diff(diff(f(x, y, z, t), x), x))*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), y))/f(x, y, z, t)^3+(2*(diff(diff(f(x, y, z, t), x), z))/f(x, y, z, t)-2*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), z))/f(x, y, z, t)^2)*delta+alpha*(2*(diff(diff(f(x, y, z, t), x), x))/f(x, y, z, t)-2*(diff(f(x, y, z, t), x))^2/f(x, y, z, t)^2)+beta*(2*(diff(diff(f(x, y, z, t), x), y))/f(x, y, z, t)-2*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), y))/f(x, y, z, t)^2)-10*(diff(diff(f(x, y, z, t), y), y))/f(x, y, z, t)+10*(diff(f(x, y, z, t), y))^2/f(x, y, z, t)^2+18*(diff(diff(f(x, y, z, t), t), x))/f(x, y, z, t)+2*(diff(diff(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x), x), x))/f(x, y, z, t)-10*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), y))/f(x, y, z, t) = 0

 (8)
> 

numer(lhs(240*(diff(diff(diff(f(x, y, z, t), x), x), x))*(diff(f(x, y, z, t), x))*(diff(diff(f(x, y, z, t), x), x))/f(x, y, z, t)^3-60*(diff(diff(f(x, y, z, t), x), x))*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), y))/f(x, y, z, t)^3-18*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), t))/f(x, y, z, t)^2-12*(diff(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x), x))*(diff(f(x, y, z, t), x))/f(x, y, z, t)^2-30*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x))*(diff(diff(f(x, y, z, t), x), x))/f(x, y, z, t)^2+10*(diff(diff(diff(f(x, y, z, t), x), x), x))*(diff(f(x, y, z, t), y))/f(x, y, z, t)^2+30*(diff(diff(diff(f(x, y, z, t), x), x), y))*(diff(f(x, y, z, t), x))/f(x, y, z, t)^2+30*(diff(diff(f(x, y, z, t), x), x))*(diff(diff(f(x, y, z, t), x), y))/f(x, y, z, t)^2-60*(diff(f(x, y, z, t), x))^2*(diff(diff(f(x, y, z, t), x), y))/f(x, y, z, t)^3+60*(diff(f(x, y, z, t), x))^3*(diff(f(x, y, z, t), y))/f(x, y, z, t)^4+60*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x))*(diff(f(x, y, z, t), x))^2/f(x, y, z, t)^3-240*(diff(diff(diff(f(x, y, z, t), x), x), x))*(diff(f(x, y, z, t), x))^3/f(x, y, z, t)^4+720*(diff(diff(f(x, y, z, t), x), x))*(diff(f(x, y, z, t), x))^4/f(x, y, z, t)^5-540*(diff(diff(f(x, y, z, t), x), x))^2*(diff(f(x, y, z, t), x))^2/f(x, y, z, t)^4+beta*(2*(diff(diff(f(x, y, z, t), x), y))/f(x, y, z, t)-2*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), y))/f(x, y, z, t)^2)-20*(diff(diff(diff(f(x, y, z, t), x), x), x))^2/f(x, y, z, t)^2+60*(diff(diff(f(x, y, z, t), x), x))^3/f(x, y, z, t)^3-240*(diff(f(x, y, z, t), x))^6/f(x, y, z, t)^6-10*(diff(diff(f(x, y, z, t), y), y))/f(x, y, z, t)+10*(diff(f(x, y, z, t), y))^2/f(x, y, z, t)^2+18*(diff(diff(f(x, y, z, t), t), x))/f(x, y, z, t)+2*(diff(diff(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x), x), x))/f(x, y, z, t)-10*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), y))/f(x, y, z, t)+(2*(diff(diff(f(x, y, z, t), x), z))/f(x, y, z, t)-2*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), z))/f(x, y, z, t)^2)*delta+alpha*(2*(diff(diff(f(x, y, z, t), x), x))/f(x, y, z, t)-2*(diff(f(x, y, z, t), x))^2/f(x, y, z, t)^2) = 0))*denom(rhs(240*(diff(diff(diff(f(x, y, z, t), x), x), x))*(diff(f(x, y, z, t), x))*(diff(diff(f(x, y, z, t), x), x))/f(x, y, z, t)^3-60*(diff(diff(f(x, y, z, t), x), x))*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), y))/f(x, y, z, t)^3-18*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), t))/f(x, y, z, t)^2-12*(diff(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x), x))*(diff(f(x, y, z, t), x))/f(x, y, z, t)^2-30*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x))*(diff(diff(f(x, y, z, t), x), x))/f(x, y, z, t)^2+10*(diff(diff(diff(f(x, y, z, t), x), x), x))*(diff(f(x, y, z, t), y))/f(x, y, z, t)^2+30*(diff(diff(diff(f(x, y, z, t), x), x), y))*(diff(f(x, y, z, t), x))/f(x, y, z, t)^2+30*(diff(diff(f(x, y, z, t), x), x))*(diff(diff(f(x, y, z, t), x), y))/f(x, y, z, t)^2-60*(diff(f(x, y, z, t), x))^2*(diff(diff(f(x, y, z, t), x), y))/f(x, y, z, t)^3+60*(diff(f(x, y, z, t), x))^3*(diff(f(x, y, z, t), y))/f(x, y, z, t)^4+60*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x))*(diff(f(x, y, z, t), x))^2/f(x, y, z, t)^3-240*(diff(diff(diff(f(x, y, z, t), x), x), x))*(diff(f(x, y, z, t), x))^3/f(x, y, z, t)^4+720*(diff(diff(f(x, y, z, t), x), x))*(diff(f(x, y, z, t), x))^4/f(x, y, z, t)^5-540*(diff(diff(f(x, y, z, t), x), x))^2*(diff(f(x, y, z, t), x))^2/f(x, y, z, t)^4+beta*(2*(diff(diff(f(x, y, z, t), x), y))/f(x, y, z, t)-2*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), y))/f(x, y, z, t)^2)-20*(diff(diff(diff(f(x, y, z, t), x), x), x))^2/f(x, y, z, t)^2+60*(diff(diff(f(x, y, z, t), x), x))^3/f(x, y, z, t)^3-240*(diff(f(x, y, z, t), x))^6/f(x, y, z, t)^6-10*(diff(diff(f(x, y, z, t), y), y))/f(x, y, z, t)+10*(diff(f(x, y, z, t), y))^2/f(x, y, z, t)^2+18*(diff(diff(f(x, y, z, t), t), x))/f(x, y, z, t)+2*(diff(diff(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x), x), x))/f(x, y, z, t)-10*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), y))/f(x, y, z, t)+(2*(diff(diff(f(x, y, z, t), x), z))/f(x, y, z, t)-2*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), z))/f(x, y, z, t)^2)*delta+alpha*(2*(diff(diff(f(x, y, z, t), x), x))/f(x, y, z, t)-2*(diff(f(x, y, z, t), x))^2/f(x, y, z, t)^2) = 0)) = numer(rhs(240*(diff(diff(diff(f(x, y, z, t), x), x), x))*(diff(f(x, y, z, t), x))*(diff(diff(f(x, y, z, t), x), x))/f(x, y, z, t)^3-60*(diff(diff(f(x, y, z, t), x), x))*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), y))/f(x, y, z, t)^3-18*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), t))/f(x, y, z, t)^2-12*(diff(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x), x))*(diff(f(x, y, z, t), x))/f(x, y, z, t)^2-30*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x))*(diff(diff(f(x, y, z, t), x), x))/f(x, y, z, t)^2+10*(diff(diff(diff(f(x, y, z, t), x), x), x))*(diff(f(x, y, z, t), y))/f(x, y, z, t)^2+30*(diff(diff(diff(f(x, y, z, t), x), x), y))*(diff(f(x, y, z, t), x))/f(x, y, z, t)^2+30*(diff(diff(f(x, y, z, t), x), x))*(diff(diff(f(x, y, z, t), x), y))/f(x, y, z, t)^2-60*(diff(f(x, y, z, t), x))^2*(diff(diff(f(x, y, z, t), x), y))/f(x, y, z, t)^3+60*(diff(f(x, y, z, t), x))^3*(diff(f(x, y, z, t), y))/f(x, y, z, t)^4+60*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x))*(diff(f(x, y, z, t), x))^2/f(x, y, z, t)^3-240*(diff(diff(diff(f(x, y, z, t), x), x), x))*(diff(f(x, y, z, t), x))^3/f(x, y, z, t)^4+720*(diff(diff(f(x, y, z, t), x), x))*(diff(f(x, y, z, t), x))^4/f(x, y, z, t)^5-540*(diff(diff(f(x, y, z, t), x), x))^2*(diff(f(x, y, z, t), x))^2/f(x, y, z, t)^4+beta*(2*(diff(diff(f(x, y, z, t), x), y))/f(x, y, z, t)-2*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), y))/f(x, y, z, t)^2)-20*(diff(diff(diff(f(x, y, z, t), x), x), x))^2/f(x, y, z, t)^2+60*(diff(diff(f(x, y, z, t), x), x))^3/f(x, y, z, t)^3-240*(diff(f(x, y, z, t), x))^6/f(x, y, z, t)^6-10*(diff(diff(f(x, y, z, t), y), y))/f(x, y, z, t)+10*(diff(f(x, y, z, t), y))^2/f(x, y, z, t)^2+18*(diff(diff(f(x, y, z, t), t), x))/f(x, y, z, t)+2*(diff(diff(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x), x), x))/f(x, y, z, t)-10*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), y))/f(x, y, z, t)+(2*(diff(diff(f(x, y, z, t), x), z))/f(x, y, z, t)-2*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), z))/f(x, y, z, t)^2)*delta+alpha*(2*(diff(diff(f(x, y, z, t), x), x))/f(x, y, z, t)-2*(diff(f(x, y, z, t), x))^2/f(x, y, z, t)^2) = 0))*denom(lhs(240*(diff(diff(diff(f(x, y, z, t), x), x), x))*(diff(f(x, y, z, t), x))*(diff(diff(f(x, y, z, t), x), x))/f(x, y, z, t)^3-60*(diff(diff(f(x, y, z, t), x), x))*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), y))/f(x, y, z, t)^3-18*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), t))/f(x, y, z, t)^2-12*(diff(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x), x))*(diff(f(x, y, z, t), x))/f(x, y, z, t)^2-30*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x))*(diff(diff(f(x, y, z, t), x), x))/f(x, y, z, t)^2+10*(diff(diff(diff(f(x, y, z, t), x), x), x))*(diff(f(x, y, z, t), y))/f(x, y, z, t)^2+30*(diff(diff(diff(f(x, y, z, t), x), x), y))*(diff(f(x, y, z, t), x))/f(x, y, z, t)^2+30*(diff(diff(f(x, y, z, t), x), x))*(diff(diff(f(x, y, z, t), x), y))/f(x, y, z, t)^2-60*(diff(f(x, y, z, t), x))^2*(diff(diff(f(x, y, z, t), x), y))/f(x, y, z, t)^3+60*(diff(f(x, y, z, t), x))^3*(diff(f(x, y, z, t), y))/f(x, y, z, t)^4+60*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x))*(diff(f(x, y, z, t), x))^2/f(x, y, z, t)^3-240*(diff(diff(diff(f(x, y, z, t), x), x), x))*(diff(f(x, y, z, t), x))^3/f(x, y, z, t)^4+720*(diff(diff(f(x, y, z, t), x), x))*(diff(f(x, y, z, t), x))^4/f(x, y, z, t)^5-540*(diff(diff(f(x, y, z, t), x), x))^2*(diff(f(x, y, z, t), x))^2/f(x, y, z, t)^4+beta*(2*(diff(diff(f(x, y, z, t), x), y))/f(x, y, z, t)-2*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), y))/f(x, y, z, t)^2)-20*(diff(diff(diff(f(x, y, z, t), x), x), x))^2/f(x, y, z, t)^2+60*(diff(diff(f(x, y, z, t), x), x))^3/f(x, y, z, t)^3-240*(diff(f(x, y, z, t), x))^6/f(x, y, z, t)^6-10*(diff(diff(f(x, y, z, t), y), y))/f(x, y, z, t)+10*(diff(f(x, y, z, t), y))^2/f(x, y, z, t)^2+18*(diff(diff(f(x, y, z, t), t), x))/f(x, y, z, t)+2*(diff(diff(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x), x), x))/f(x, y, z, t)-10*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), y))/f(x, y, z, t)+(2*(diff(diff(f(x, y, z, t), x), z))/f(x, y, z, t)-2*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), z))/f(x, y, z, t)^2)*delta+alpha*(2*(diff(diff(f(x, y, z, t), x), x))/f(x, y, z, t)-2*(diff(f(x, y, z, t), x))^2/f(x, y, z, t)^2) = 0))

2*f(x, y, z, t)^5*(diff(diff(f(x, y, z, t), x), x))*alpha+2*f(x, y, z, t)^5*(diff(diff(f(x, y, z, t), x), y))*beta+720*(diff(diff(f(x, y, z, t), x), x))*(diff(f(x, y, z, t), x))^4*f(x, y, z, t)-240*(diff(diff(diff(f(x, y, z, t), x), x), x))*(diff(f(x, y, z, t), x))^3*f(x, y, z, t)^2+60*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x))*(diff(f(x, y, z, t), x))^2*f(x, y, z, t)^3-60*(diff(f(x, y, z, t), x))^2*(diff(diff(f(x, y, z, t), x), y))*f(x, y, z, t)^3-540*(diff(diff(f(x, y, z, t), x), x))^2*(diff(f(x, y, z, t), x))^2*f(x, y, z, t)^2-18*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), t))*f(x, y, z, t)^4-12*(diff(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x), x))*(diff(f(x, y, z, t), x))*f(x, y, z, t)^4+30*(diff(diff(diff(f(x, y, z, t), x), x), y))*(diff(f(x, y, z, t), x))*f(x, y, z, t)^4-30*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x))*(diff(diff(f(x, y, z, t), x), x))*f(x, y, z, t)^4+30*(diff(diff(f(x, y, z, t), x), x))*(diff(diff(f(x, y, z, t), x), y))*f(x, y, z, t)^4+10*(diff(diff(diff(f(x, y, z, t), x), x), x))*(diff(f(x, y, z, t), y))*f(x, y, z, t)^4+2*(diff(diff(f(x, y, z, t), x), z))*delta*f(x, y, z, t)^5+60*(diff(f(x, y, z, t), x))^3*(diff(f(x, y, z, t), y))*f(x, y, z, t)^2-2*(diff(f(x, y, z, t), x))^2*alpha*f(x, y, z, t)^4-240*(diff(f(x, y, z, t), x))^6-2*(diff(f(x, y, z, t), x))*f(x, y, z, t)^4*(diff(f(x, y, z, t), y))*beta-2*(diff(f(x, y, z, t), x))*f(x, y, z, t)^4*(diff(f(x, y, z, t), z))*delta+240*(diff(diff(diff(f(x, y, z, t), x), x), x))*(diff(f(x, y, z, t), x))*(diff(diff(f(x, y, z, t), x), x))*f(x, y, z, t)^3-60*(diff(diff(f(x, y, z, t), x), x))*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), y))*f(x, y, z, t)^3+2*(diff(diff(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x), x), x))*f(x, y, z, t)^5-10*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), y))*f(x, y, z, t)^5-20*(diff(diff(diff(f(x, y, z, t), x), x), x))^2*f(x, y, z, t)^4+60*(diff(diff(f(x, y, z, t), x), x))^3*f(x, y, z, t)^3+18*(diff(diff(f(x, y, z, t), t), x))*f(x, y, z, t)^5-10*(diff(diff(f(x, y, z, t), y), y))*f(x, y, z, t)^5+10*(diff(f(x, y, z, t), y))^2*f(x, y, z, t)^4 = 0

 (9)
> 

simplify(2*(diff(diff(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x), x), x))*f(x, y, z, t)^5-10*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), y))*f(x, y, z, t)^5-20*(diff(diff(diff(f(x, y, z, t), x), x), x))^2*f(x, y, z, t)^4+60*(diff(diff(f(x, y, z, t), x), x))^3*f(x, y, z, t)^3+18*(diff(diff(f(x, y, z, t), t), x))*f(x, y, z, t)^5-10*(diff(diff(f(x, y, z, t), y), y))*f(x, y, z, t)^5+10*(diff(f(x, y, z, t), y))^2*f(x, y, z, t)^4+2*f(x, y, z, t)^5*(diff(diff(f(x, y, z, t), x), y))*beta+720*(diff(diff(f(x, y, z, t), x), x))*(diff(f(x, y, z, t), x))^4*f(x, y, z, t)-240*(diff(diff(diff(f(x, y, z, t), x), x), x))*(diff(f(x, y, z, t), x))^3*f(x, y, z, t)^2+60*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x))*(diff(f(x, y, z, t), x))^2*f(x, y, z, t)^3-60*(diff(f(x, y, z, t), x))^2*(diff(diff(f(x, y, z, t), x), y))*f(x, y, z, t)^3-540*(diff(diff(f(x, y, z, t), x), x))^2*(diff(f(x, y, z, t), x))^2*f(x, y, z, t)^2-18*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), t))*f(x, y, z, t)^4-12*(diff(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x), x))*(diff(f(x, y, z, t), x))*f(x, y, z, t)^4+30*(diff(diff(diff(f(x, y, z, t), x), x), y))*(diff(f(x, y, z, t), x))*f(x, y, z, t)^4-30*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x))*(diff(diff(f(x, y, z, t), x), x))*f(x, y, z, t)^4+30*(diff(diff(f(x, y, z, t), x), x))*(diff(diff(f(x, y, z, t), x), y))*f(x, y, z, t)^4+10*(diff(diff(diff(f(x, y, z, t), x), x), x))*(diff(f(x, y, z, t), y))*f(x, y, z, t)^4+2*(diff(diff(f(x, y, z, t), x), z))*delta*f(x, y, z, t)^5+60*(diff(f(x, y, z, t), x))^3*(diff(f(x, y, z, t), y))*f(x, y, z, t)^2-2*(diff(f(x, y, z, t), x))^2*alpha*f(x, y, z, t)^4+2*f(x, y, z, t)^5*(diff(diff(f(x, y, z, t), x), x))*alpha-240*(diff(f(x, y, z, t), x))^6-2*(diff(f(x, y, z, t), x))*f(x, y, z, t)^4*(diff(f(x, y, z, t), y))*beta-2*(diff(f(x, y, z, t), x))*f(x, y, z, t)^4*(diff(f(x, y, z, t), z))*delta+240*(diff(diff(diff(f(x, y, z, t), x), x), x))*(diff(f(x, y, z, t), x))*(diff(diff(f(x, y, z, t), x), x))*f(x, y, z, t)^3-60*(diff(diff(f(x, y, z, t), x), x))*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), y))*f(x, y, z, t)^3 = 0)

2*(diff(diff(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x), x), x))*f(x, y, z, t)^5-12*(diff(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x), x))*(diff(f(x, y, z, t), x))*f(x, y, z, t)^4+30*(2*(diff(f(x, y, z, t), x))^2*f(x, y, z, t)^3-f(x, y, z, t)^4*(diff(diff(f(x, y, z, t), x), x)))*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x))-10*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), y))*f(x, y, z, t)^5-20*(diff(diff(diff(f(x, y, z, t), x), x), x))^2*f(x, y, z, t)^4+10*(-24*(diff(f(x, y, z, t), x))^3*f(x, y, z, t)^2+24*(diff(f(x, y, z, t), x))*f(x, y, z, t)^3*(diff(diff(f(x, y, z, t), x), x))+(diff(f(x, y, z, t), y))*f(x, y, z, t)^4)*(diff(diff(diff(f(x, y, z, t), x), x), x))+30*(diff(diff(diff(f(x, y, z, t), x), x), y))*(diff(f(x, y, z, t), x))*f(x, y, z, t)^4+60*(diff(diff(f(x, y, z, t), x), x))^3*f(x, y, z, t)^3-540*(diff(diff(f(x, y, z, t), x), x))^2*(diff(f(x, y, z, t), x))^2*f(x, y, z, t)^2+2*(alpha*f(x, y, z, t)^5+360*(diff(f(x, y, z, t), x))^4*f(x, y, z, t)-30*(diff(f(x, y, z, t), y))*f(x, y, z, t)^3*(diff(f(x, y, z, t), x))+15*f(x, y, z, t)^4*(diff(diff(f(x, y, z, t), x), y)))*(diff(diff(f(x, y, z, t), x), x))+2*(beta*f(x, y, z, t)^5-30*(diff(f(x, y, z, t), x))^2*f(x, y, z, t)^3)*(diff(diff(f(x, y, z, t), x), y))+18*(diff(diff(f(x, y, z, t), t), x))*f(x, y, z, t)^5+2*(diff(diff(f(x, y, z, t), x), z))*delta*f(x, y, z, t)^5-10*(diff(diff(f(x, y, z, t), y), y))*f(x, y, z, t)^5-240*(diff(f(x, y, z, t), x))^6+60*(diff(f(x, y, z, t), x))^3*(diff(f(x, y, z, t), y))*f(x, y, z, t)^2-2*(diff(f(x, y, z, t), x))^2*alpha*f(x, y, z, t)^4-2*f(x, y, z, t)^4*(beta*(diff(f(x, y, z, t), y))+(diff(f(x, y, z, t), z))*delta+9*(diff(f(x, y, z, t), t)))*(diff(f(x, y, z, t), x))+10*(diff(f(x, y, z, t), y))^2*f(x, y, z, t)^4 = 0

 (10)
> 

F1 := %*(1/2)

(diff(diff(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x), x), x))*f(x, y, z, t)^5-6*(diff(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x), x))*(diff(f(x, y, z, t), x))*f(x, y, z, t)^4+15*(2*(diff(f(x, y, z, t), x))^2*f(x, y, z, t)^3-f(x, y, z, t)^4*(diff(diff(f(x, y, z, t), x), x)))*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x))-5*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), y))*f(x, y, z, t)^5-10*(diff(diff(diff(f(x, y, z, t), x), x), x))^2*f(x, y, z, t)^4+5*(-24*(diff(f(x, y, z, t), x))^3*f(x, y, z, t)^2+24*(diff(f(x, y, z, t), x))*f(x, y, z, t)^3*(diff(diff(f(x, y, z, t), x), x))+(diff(f(x, y, z, t), y))*f(x, y, z, t)^4)*(diff(diff(diff(f(x, y, z, t), x), x), x))+15*(diff(diff(diff(f(x, y, z, t), x), x), y))*(diff(f(x, y, z, t), x))*f(x, y, z, t)^4+30*(diff(diff(f(x, y, z, t), x), x))^3*f(x, y, z, t)^3-270*(diff(diff(f(x, y, z, t), x), x))^2*(diff(f(x, y, z, t), x))^2*f(x, y, z, t)^2+(alpha*f(x, y, z, t)^5+360*(diff(f(x, y, z, t), x))^4*f(x, y, z, t)-30*(diff(f(x, y, z, t), y))*f(x, y, z, t)^3*(diff(f(x, y, z, t), x))+15*f(x, y, z, t)^4*(diff(diff(f(x, y, z, t), x), y)))*(diff(diff(f(x, y, z, t), x), x))+(beta*f(x, y, z, t)^5-30*(diff(f(x, y, z, t), x))^2*f(x, y, z, t)^3)*(diff(diff(f(x, y, z, t), x), y))+9*(diff(diff(f(x, y, z, t), t), x))*f(x, y, z, t)^5+(diff(diff(f(x, y, z, t), x), z))*delta*f(x, y, z, t)^5-5*(diff(diff(f(x, y, z, t), y), y))*f(x, y, z, t)^5-120*(diff(f(x, y, z, t), x))^6+30*(diff(f(x, y, z, t), x))^3*(diff(f(x, y, z, t), y))*f(x, y, z, t)^2-(diff(f(x, y, z, t), x))^2*alpha*f(x, y, z, t)^4-f(x, y, z, t)^4*(beta*(diff(f(x, y, z, t), y))+(diff(f(x, y, z, t), z))*delta+9*(diff(f(x, y, z, t), t)))*(diff(f(x, y, z, t), x))+5*(diff(f(x, y, z, t), y))^2*f(x, y, z, t)^4 = 0

 (11)
> 

collect(F1, {alpha, beta, f(x, y, z, t)})

(-f(x, y, z, t)^4*(diff(f(x, y, z, t), x))^2+f(x, y, z, t)^5*(diff(diff(f(x, y, z, t), x), x)))*alpha+(-(diff(f(x, y, z, t), y))*f(x, y, z, t)^4*(diff(f(x, y, z, t), x))+f(x, y, z, t)^5*(diff(diff(f(x, y, z, t), x), y)))*beta+((diff(diff(f(x, y, z, t), x), z))*delta+diff(diff(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x), x), x)-5*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), y))+9*(diff(diff(f(x, y, z, t), t), x))-5*(diff(diff(f(x, y, z, t), y), y)))*f(x, y, z, t)^5+(-6*(diff(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x), x))*(diff(f(x, y, z, t), x))-15*(diff(diff(f(x, y, z, t), x), x))*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x))-10*(diff(diff(diff(f(x, y, z, t), x), x), x))^2+5*(diff(f(x, y, z, t), y))*(diff(diff(diff(f(x, y, z, t), x), x), x))+15*(diff(diff(diff(f(x, y, z, t), x), x), y))*(diff(f(x, y, z, t), x))+15*(diff(diff(f(x, y, z, t), x), y))*(diff(diff(f(x, y, z, t), x), x))-((diff(f(x, y, z, t), z))*delta+9*(diff(f(x, y, z, t), t)))*(diff(f(x, y, z, t), x))+5*(diff(f(x, y, z, t), y))^2)*f(x, y, z, t)^4+(30*(diff(f(x, y, z, t), x))^2*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x))-30*(diff(f(x, y, z, t), x))^2*(diff(diff(f(x, y, z, t), x), y))+120*(diff(f(x, y, z, t), x))*(diff(diff(f(x, y, z, t), x), x))*(diff(diff(diff(f(x, y, z, t), x), x), x))-30*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), y))*(diff(diff(f(x, y, z, t), x), x))+30*(diff(diff(f(x, y, z, t), x), x))^3)*f(x, y, z, t)^3+(-120*(diff(f(x, y, z, t), x))^3*(diff(diff(diff(f(x, y, z, t), x), x), x))+30*(diff(f(x, y, z, t), x))^3*(diff(f(x, y, z, t), y))-270*(diff(diff(f(x, y, z, t), x), x))^2*(diff(f(x, y, z, t), x))^2)*f(x, y, z, t)^2+360*(diff(diff(f(x, y, z, t), x), x))*(diff(f(x, y, z, t), x))^4*f(x, y, z, t)-120*(diff(f(x, y, z, t), x))^6 = 0

 (12)
> 

NULL
> 

T := f(x, y, z, t) = g(x, y, z, t)^2+h(x, y, z, t)^2+a[11]
> 

T1 := g(x, y, z, t) = t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5]
> 

T2 := h(x, y, z, t) = t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10]
> 

L2 := subs({T1, T2}, T)

f(x, y, z, t) = (t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])^2+(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])^2+a[11]

 (13)
> 

L3 := eval(F1, L2)

30*(2*a[1]^2+2*a[6]^2)^3*((t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])^2+(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])^2+a[11])^3-270*(2*a[1]^2+2*a[6]^2)^2*(2*(t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])*a[1]+2*(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])*a[6])^2*((t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])^2+(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])^2+a[11])^2+(alpha*((t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])^2+(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])^2+a[11])^5+360*(2*(t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])*a[1]+2*(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])*a[6])^4*((t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])^2+(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])^2+a[11])-30*(2*(t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])*a[2]+2*(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])*a[7])*((t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])^2+(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])^2+a[11])^3*(2*(t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])*a[1]+2*(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])*a[6])+15*((t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])^2+(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])^2+a[11])^4*(2*a[1]*a[2]+2*a[6]*a[7]))*(2*a[1]^2+2*a[6]^2)+(beta*((t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])^2+(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])^2+a[11])^5-30*(2*(t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])*a[1]+2*(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])*a[6])^2*((t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])^2+(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])^2+a[11])^3)*(2*a[1]*a[2]+2*a[6]*a[7])+9*(2*a[1]*a[4]+2*a[6]*a[9])*((t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])^2+(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])^2+a[11])^5+(2*a[1]*a[3]+2*a[6]*a[8])*delta*((t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])^2+(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])^2+a[11])^5-5*(2*a[2]^2+2*a[7]^2)*((t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])^2+(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])^2+a[11])^5-120*(2*(t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])*a[1]+2*(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])*a[6])^6+30*(2*(t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])*a[1]+2*(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])*a[6])^3*(2*(t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])*a[2]+2*(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])*a[7])*((t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])^2+(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])^2+a[11])^2-(2*(t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])*a[1]+2*(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])*a[6])^2*alpha*((t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])^2+(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])^2+a[11])^4-((t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])^2+(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])^2+a[11])^4*(beta*(2*(t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])*a[2]+2*(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])*a[7])+(2*(t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])*a[3]+2*(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])*a[8])*delta+18*(t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])*a[4]+18*(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])*a[9])*(2*(t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])*a[1]+2*(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])*a[6])+5*(2*(t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])*a[2]+2*(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])*a[7])^2*((t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])^2+(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])^2+a[11])^4 = 0

 (14)
> 

L4 := collect(L3, [x, y, z, t], 'distributed')

Warning,  computation interrupted

  

` `

 (15)

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