Variables of the form z_xxxx are *integer* variables created by Maxima,
which attempts to give you *also* the complex roots, if any, thus ignoring
the assumptions on x, y and l. Note that :
sage: solve(FOC[0], x)
---------------------------------------------------------------------------
[ Snip… ]
TypeError: Computation failed since Maxima requested additional constraints;
using the 'assume' command before evaluation *may* help (example of legal
syntax is 'assume(l>0)', see `assume?` for more details)
Is l positive, negative or zero?
sage: with assuming(l>0): print(solve(FOC[0], x))
[
x == (l*p_x)^(1/a)
]
sage: with assuming(l<0): print(solve(FOC[0], x))
[
x^a == l*p_x
]
sage: with assuming(l<0): print(solve(FOC[0], x, to_poly_solve=True))
[x == (l*p_x)^(1/a)*e^(2*I*pi*z4353/a)]
Interestingly:
sage: solve([FOC[0]==0,FOC[1]==0,FOC[2]==0],x,y,l)
[[l == (1/p_y), x == (p_x/p_y)^(1/a)*e^(2*I*pi*z3540/a), y ==
-(p_x*(p_x/p_y)^(1/a)*e^(2*I*pi*z3540/a) - R)/p_y]]
sage: solve([FOC[0]==0,FOC[1]==0,FOC[2]==0],x,y,l, algorithm="sympy")
[{x: (p_x/p_y)^(1/a), l: 1/p_y, y: -(p_x*(p_x/p_y)^(1/a) - R)/p_y}]
sage: solve([FOC[0]==0,FOC[1]==0,FOC[2]==0],x,y,l, algorithm="fricas")
[[l == (1/p_y), x == (p_x/p_y)^(1/a)*e^(2*I*pi*z3891/a), y ==
-(p_x*(p_x/p_y)^(1/a)*e^(2*I*pi*z3891/a) - R)/p_y]]
sage: giac.solve(giac(FOC),giac([x,y,l])).sage()
[[(p_x/p_y)^(1/a), -(p_x*(p_x/p_y)^(1/a) - R)/p_y, 1/p_y]]
HTH,
Le dimanche 28 novembre 2021 à 22:13:12 UTC+1, cyrille piatecki a écrit :
> On my computer the solution of
>
> var('a x y p_x p_y D Rev R l')
> assume(a,'real')
> assume(x,'real')
> assume(y,'real')
> assume(p_x,'real')
> assume(p_y,'real')
> assume(D,'real')
> assume(Rev,'real')
> assume(R,'real')
> assume(l,'real')
> assume(a<1)
> assume(a>0)
> assume(p_x>0)
> assume(p_y>0)
> assume(R>0)
> U =(1/(a+1))*x^(a+1)+y
> show(LatexExpr(r'\text{La fonction d}^\prime\text{utilité est }U(x,y) =
> '),U)
> D= x*p_x + y*p_y
> show(LatexExpr(r'\text{La Dépense } D = '),D)
> Rev= R
> show(LatexExpr(r'\text{Le Revenu } Rev = '),R)
> L=U+l*(Rev-D)
> show(LatexExpr(r'\text{Le lagrangien est } \mathcal{L}(x, y, λ) = '),L)
> FOC = [diff(L,x),diff(L,y),diff(L,l)]
> show(LatexExpr(r'\text{Les condition du premier ordre sont }
> \left\{\begin{array}{c}\mathcal{L}_x= 0\\\mathcal{L}_y= 0\\\mathcal{L}_λ=
> 0\end{array}\right. '))
> show(LatexExpr(r'\text{soit }'))
> show(LatexExpr(r'\mathcal{L}_x= 0 \Longleftrightarrow '),FOC[0]==0)
> show(LatexExpr(r'\mathcal{L}_y= 0 \Longleftrightarrow '),FOC[1]==0)
> show(LatexExpr(r'\mathcal{L}_λ= 0 \Longleftrightarrow '),FOC[2]==0)
> sol=solve([FOC[0]==0,FOC[1]==0,FOC[2]==0],x,y,l)
> show(sol)
>
> Is nearly correct, but an extra complex exponential term multiplies $x$
> and then modifies $y$. Even as an element I do not understand its form :
>
> $e^{(2iπz_{5797}a)}$
>
> Could some one explain why ?
>
>
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