Slove function return empty solutions
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Hello, I'm trying to solve the attached syntax, but the aolve function return empty solutions. Please help.
syms V_1 V_2 x_1 x_2 r
pi1 = (V_1) * (x_1^r/(x_1^r+x_2^r)) - x_1
pi2 = (V_2) * (x_2^r/(x_1^r+x_2^r)) - x_2
dpi1dx = diff(pi1, x_1)
dpi2dx = diff(pi2, x_2)
s = solve(dpi1dx==0, dpi2dx==0, x_1, x_2)
2 comentarios
Respuestas (2)
Walter Roberson
el 16 de Mzo. de 2023
Use dsolve for differential equations
20 comentarios
Walter Roberson
el 21 de Mzo. de 2023
The problem is not solveable for most r .
For example for r = 3/2 then the solutions are
RootOf(4*Z^3*x_2^(3/2) + 2*Z^6 - 3*Z*x_2^(3/2)*V_1 + 2*x_2^3,Z)^2
which is the set of Z such that the expression 4*etc becomes 0. But notice the Z^6 part -- so you would need the closed-form solution for a degree 6 polynomial, and such solutions only exist if the expression can be factored into polynomials of degree 4 or lower.
If r = N/4 for odd integer N, then you need to solve something of degree either 2*N+4 (for small N) or degree 2*N (starting at N = 5). r = 1/5 and r = 3/5 are tractable (but long!!), the other N/5 are not tractable.
Roy
el 21 de Mzo. de 2023
3 comentarios
Walter Roberson
el 22 de Mzo. de 2023
If you add the assumption of positive then they do resolve to 0
syms V_1 V_2 x_1 x_2 r positive
pi1 = (V_1) * (x_1^r/(x_1^r+x_2^r)) - x_1
pi2 = (V_2) * (x_2^r/(x_1^r+x_2^r)) - x_2
dpi1dx = diff(pi1, x_1)
dpi2dx = diff(pi2, x_2)
simplify(subs(dpi1dx,[x_1 x_2],[V_2*(r*(V_2/V_1)^(r-1))/(1+(V_2/V_1)^r)^2,V_1*(r*(V_1/V_2)^(r-1))/(1+(V_1/V_2)^r)^2]))
simplify(subs(dpi2dx,[x_1 x_2],[V_2*(r*(V_2/V_1)^(r-1))/(1+(V_2/V_1)^r)^2,V_1*(r*(V_1/V_2)^(r-1))/(1+(V_1/V_2)^r)^2]))
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