System of 2 nonlinear equations: no result

Questions
1

Why I can not get the answers of x, y, what’s wrong with it? Pls guide me, Thanks in advance.

1.png
1.png1492×174 14.8 KB

2

Please post CODE not the image or attach the notebook.

3

Added code and the .nb file. Thanks.

Solve[{(? ? + ? ? y) (p - ? x^2 - (1 - ?) y^2 - b - d) + 
(a - k p + ? (? x + (1 - ?) y + ? ? y)) (-2 ? x) == 0, 
(? (1 - ?) + ? ? x) (p - ? x^2 - (1 - ?) y^2 - b - d) + 
(a - k p + ? (? x + (1 - ?) y + ? ? y)) (-2 (1 - ?) y) == 0}, {x, y}]
4

Your system is of high degree with symbolic coefficients. A full symbolic solution may be too complicated. If you lower your expectations, you get quick explicit solutions for specific numeric values of the parameters, for example

With[{\[Lambda] = 1, \[Alpha] = -1, \[Mu] = 1, p = 1, \[Theta] = 1, 
  b = 1, d = 1, a = 1, k = 1},
 Solve[{(\[Lambda] \[Alpha] + \[Lambda] \[Mu] y) (p - \[Theta] x^2 - \
(1 - \[Theta]) y^2 - b - d) + (a - 
        k p + \[Lambda] (\[Alpha] x + (1 - \[Alpha]) y + \[Lambda] \
\[Mu] y)) (-2 \[Theta] x) == 
    0, (\[Lambda] (1 - \[Alpha]) + \[Lambda] \[Mu] x) (p - \[Theta] \
x^2 - (1 - \[Theta]) y^2 - b - d) + (a - 
        k p + \[Lambda] (\[Alpha] x + (1 - \[Alpha]) y + \[Lambda] \
\[Mu] y)) (-2 (1 - \[Theta]) y) == 0}, {x, y}]]

You can go interactive too:

Manipulate[
 ContourPlot[{(\[Lambda] \[Alpha] + \[Lambda] \[Mu] y) (p - \[Theta] \
x^2 - (1 - \[Theta]) y^2 - b - d) + (a - 
        k p + \[Lambda] (\[Alpha] x + (1 - \[Alpha]) y + \[Lambda] \
\[Mu] y)) (-2 \[Theta] x) == 
    0, (\[Lambda] (1 - \[Alpha]) + \[Lambda] \[Mu] x) (p - \[Theta] \
x^2 - (1 - \[Theta]) y^2 - b - d) + (a - 
        k p + \[Lambda] (\[Alpha] x + (1 - \[Alpha]) y + \[Lambda] \
\[Mu] y)) (-2 (1 - \[Theta]) y) == 0}, {x, -1, 1}, {y, -1, 1}],
 {\[Lambda], -1, 1}, {\[Alpha], -1, 1}, {\[Mu], -1, 1}, {p, -1, 
  1}, {\[Theta], -1, 1}, {b, -1, 1}, {d, -1, 1}, {a, -1, 1}, {k, -1, 
  1}]
5

Clear[“Global`*”]
eq1 = FullSimplify[(? ? + ? ? y) (p - ? x^2 - (1 - ?) y^2 - b - d) +
(a - k p + ? (? x + (1 - ?) y + ? ? y)) (-2 ? x) == 0]
eq2 = FullSimplify[(? (1 - ?) + ? ? x) (p - ? x^2 - (1 - ?) y^2 - b - d) +
(a - k p + ? (? x + (1 - ?) y + ? ? y)) (-2 (1 - ?) y) == 0]
for Y:

Y = Solve[Eliminate[{eq1, eq2}, x], y]

for X:

X = Solve[Eliminate[{eq1, eq2}, y], x]
6

In[4]:= Solve[Join[eq1, eq2], {x, y}]

Out[4]= {{x -> (
   1/(\[Alpha] \[Lambda]))(-a + k p + (
     a \[Theta] \[Lambda]^2)/(\[Alpha]^2 \[Lambda]^2 + \[Theta] \
\[Lambda]^2 - 2 \[Alpha] \[Theta] \[Lambda]^2 + 
      2 \[Theta] \[Lambda]^3 \[Mu] - 
      2 \[Alpha] \[Theta] \[Lambda]^3 \[Mu] + \[Theta] \[Lambda]^4 \
\[Mu]^2) - (
     k p \[Theta] \[Lambda]^2)/(\[Alpha]^2 \[Lambda]^2 + \[Theta] \
\[Lambda]^2 - 2 \[Alpha] \[Theta] \[Lambda]^2 + 
      2 \[Theta] \[Lambda]^3 \[Mu] - 
      2 \[Alpha] \[Theta] \[Lambda]^3 \[Mu] + \[Theta] \[Lambda]^4 \
\[Mu]^2) - (
     2 a \[Alpha] \[Theta] \[Lambda]^2)/(\[Alpha]^2 \[Lambda]^2 + \
\[Theta] \[Lambda]^2 - 2 \[Alpha] \[Theta] \[Lambda]^2 + 
      2 \[Theta] \[Lambda]^3 \[Mu] - 
      2 \[Alpha] \[Theta] \[Lambda]^3 \[Mu] + \[Theta] \[Lambda]^4 \
\[Mu]^2) + (
     2 k p \[Alpha] \[Theta] \[Lambda]^2)/(\[Alpha]^2 \[Lambda]^2 + \
\[Theta] \[Lambda]^2 - 2 \[Alpha] \[Theta] \[Lambda]^2 + 
      2 \[Theta] \[Lambda]^3 \[Mu] - 
      2 \[Alpha] \[Theta] \[Lambda]^3 \[Mu] + \[Theta] \[Lambda]^4 \
\[Mu]^2) + (
     a \[Alpha]^2 \[Theta] \[Lambda]^2)/(\[Alpha]^2 \[Lambda]^2 + \
\[Theta] \[Lambda]^2 - 2 \[Alpha] \[Theta] \[Lambda]^2 + 
      2 \[Theta] \[Lambda]^3 \[Mu] - 
      2 \[Alpha] \[Theta] \[Lambda]^3 \[Mu] + \[Theta] \[Lambda]^4 \
\[Mu]^2) - (
     k p \[Alpha]^2 \[Theta] \[Lambda]^2)/(\[Alpha]^2 \[Lambda]^2 + \
\[Theta] \[Lambda]^2 - 2 \[Alpha] \[Theta] \[Lambda]^2 + 
      2 \[Theta] \[Lambda]^3 \[Mu] - 
      2 \[Alpha] \[Theta] \[Lambda]^3 \[Mu] + \[Theta] \[Lambda]^4 \
\[Mu]^2) + (
     2 a \[Theta] \[Lambda]^3 \[Mu])/(\[Alpha]^2 \[Lambda]^2 + \
\[Theta] \[Lambda]^2 - 2 \[Alpha] \[Theta] \[Lambda]^2 + 
      2 \[Theta] \[Lambda]^3 \[Mu] - 
      2 \[Alpha] \[Theta] \[Lambda]^3 \[Mu] + \[Theta] \[Lambda]^4 \
\[Mu]^2) - (
     2 k p \[Theta] \[Lambda]^3 \[Mu])/(\[Alpha]^2 \[Lambda]^2 + \
\[Theta] \[Lambda]^2 - 2 \[Alpha] \[Theta] \[Lambda]^2 + 
      2 \[Theta] \[Lambda]^3 \[Mu] - 
      2 \[Alpha] \[Theta] \[Lambda]^3 \[Mu] + \[Theta] \[Lambda]^4 \
\[Mu]^2) - (
     2 a \[Alpha] \[Theta] \[Lambda]^3 \[Mu])/(\[Alpha]^2 \[Lambda]^2 \
+ \[Theta] \[Lambda]^2 - 2 \[Alpha] \[Theta] \[Lambda]^2 + 
      2 \[Theta] \[Lambda]^3 \[Mu] - 
      2 \[Alpha] \[Theta] \[Lambda]^3 \[Mu] + \[Theta] \[Lambda]^4 \
\[Mu]^2) + (
     2 k p \[Alpha] \[Theta] \[Lambda]^3 \[Mu])/(\[Alpha]^2 \
\[Lambda]^2 + \[Theta] \[Lambda]^2 - 
      2 \[Alpha] \[Theta] \[Lambda]^2 + 
      2 \[Theta] \[Lambda]^3 \[Mu] - 
      2 \[Alpha] \[Theta] \[Lambda]^3 \[Mu] + \[Theta] \[Lambda]^4 \
\[Mu]^2) + (
     a \[Theta] \[Lambda]^4 \[Mu]^2)/(\[Alpha]^2 \[Lambda]^2 + \
\[Theta] \[Lambda]^2 - 2 \[Alpha] \[Theta] \[Lambda]^2 + 
      2 \[Theta] \[Lambda]^3 \[Mu] - 
      2 \[Alpha] \[Theta] \[Lambda]^3 \[Mu] + \[Theta] \[Lambda]^4 \
\[Mu]^2) - (
     k p \[Theta] \[Lambda]^4 \[Mu]^2)/(\[Alpha]^2 \[Lambda]^2 + \
\[Theta] \[Lambda]^2 - 2 \[Alpha] \[Theta] \[Lambda]^2 + 
      2 \[Theta] \[Lambda]^3 \[Mu] - 
      2 \[Alpha] \[Theta] \[Lambda]^3 \[Mu] + \[Theta] \[Lambda]^4 \
\[Mu]^2) + (\[Lambda] \[Sqrt]((2 a \[Theta] \[Lambda] - 
             2 k p \[Theta] \[Lambda] - 
             2 a \[Alpha] \[Theta] \[Lambda] + 
             2 k p \[Alpha] \[Theta] \[Lambda] + 
             2 a \[Theta] \[Lambda]^2 \[Mu] - 
             2 k p \[Theta] \[Lambda]^2 \[Mu])^2 - 
           4 (a^2 \[Theta] - 2 a k p \[Theta] + k^2 p^2 \[Theta] + 
              b \[Alpha]^2 \[Lambda]^2 + d \[Alpha]^2 \[Lambda]^2 - 
              p \[Alpha]^2 \[Lambda]^2) (\[Alpha]^2 \[Lambda]^2 + \
\[Theta] \[Lambda]^2 - 2 \[Alpha] \[Theta] \[Lambda]^2 + 
              2 \[Theta] \[Lambda]^3 \[Mu] - 
              2 \[Alpha] \[Theta] \[Lambda]^3 \[Mu] + \[Theta] \
\[Lambda]^4 \[Mu]^2)))/(2 (\[Alpha]^2 \[Lambda]^2 + \[Theta] \
\[Lambda]^2 - 2 \[Alpha] \[Theta] \[Lambda]^2 + 
          2 \[Theta] \[Lambda]^3 \[Mu] - 
          2 \[Alpha] \[Theta] \[Lambda]^3 \[Mu] + \[Theta] \
\[Lambda]^4 \[Mu]^2)) - (\[Alpha] \[Lambda] \[Sqrt]((2 a \[Theta] \
\[Lambda] - 2 k p \[Theta] \[Lambda] - 
             2 a \[Alpha] \[Theta] \[Lambda] + 
             2 k p \[Alpha] \[Theta] \[Lambda] + 
             2 a \[Theta] \[Lambda]^2 \[Mu] - 
             2 k p \[Theta] \[Lambda]^2 \[Mu])^2 - 
           
           4 (a^2 \[Theta] - 2 a k p \[Theta] + k^2 p^2 \[Theta] + 
              b \[Alpha]^2 \[Lambda]^2 + d \[Alpha]^2 \[Lambda]^2 - 
              p \[Alpha]^2 \[Lambda]^2) (\[Alpha]^2 \[Lambda]^2 + \
\[Theta] \[Lambda]^2 - 2 \[Alpha] \[Theta] \[Lambda]^2 + 
              2 \[Theta] \[Lambda]^3 \[Mu] - 
              2 \[Alpha] \[Theta] \[Lambda]^3 \[Mu] + \[Theta] \
\[Lambda]^4 \[Mu]^2)))/(2 (\[Alpha]^2 \[Lambda]^2 + \[Theta] \
\[Lambda]^2 - 2 \[Alpha] \[Theta] \[Lambda]^2 + 
          2 \[Theta] \[Lambda]^3 \[Mu] - 
          2 \[Alpha] \[Theta] \[Lambda]^3 \[Mu] + \[Theta] \
\[Lambda]^4 \[Mu]^2)) + (\[Lambda]^2 \[Mu] \[Sqrt]((2 a \[Theta] \
\[Lambda] - 2 k p \[Theta] \[Lambda] - 
             2 a \[Alpha] \[Theta] \[Lambda] + 
             2 k p \[Alpha] \[Theta] \[Lambda] + 
             2 a \[Theta] \[Lambda]^2 \[Mu] - 
             2 k p \[Theta] \[Lambda]^2 \[Mu])^2 - 
           4 (a^2 \[Theta] - 2 a k p \[Theta] + k^2 p^2 \[Theta] + 
              b \[Alpha]^2 \[Lambda]^2 + d \[Alpha]^2 \[Lambda]^2 - 
              p \[Alpha]^2 \[Lambda]^2) (\[Alpha]^2 \[Lambda]^2 + \
\[Theta] \[Lambda]^2 - 2 \[Alpha] \[Theta] \[Lambda]^2 + 
              2 \[Theta] \[Lambda]^3 \[Mu] - 
              2 \[Alpha] \[Theta] \[Lambda]^3 \[Mu] + \[Theta] \
\[Lambda]^4 \[Mu]^2)))/(2 (\[Alpha]^2 \[Lambda]^2 + \[Theta] \
\[Lambda]^2 - 2 \[Alpha] \[Theta] \[Lambda]^2 + 
          2 \[Theta] \[Lambda]^3 \[Mu] - 
          2 \[Alpha] \[Theta] \[Lambda]^3 \[Mu] + \[Theta] \
\[Lambda]^4 \[Mu]^2))), 
  y -> (-2 a \[Theta] \[Lambda] + 2 k p \[Theta] \[Lambda] + 
      2 a \[Alpha] \[Theta] \[Lambda] - 
      2 k p \[Alpha] \[Theta] \[Lambda] - 
      2 a \[Theta] \[Lambda]^2 \[Mu] + 
      2 k p \[Theta] \[Lambda]^2 \[Mu] - \[Sqrt]((2 a \[Theta] \
\[Lambda] - 2 k p \[Theta] \[Lambda] - 
           2 a \[Alpha] \[Theta] \[Lambda] + 
           2 k p \[Alpha] \[Theta] \[Lambda] + 
           2 a \[Theta] \[Lambda]^2 \[Mu] - 
           2 k p \[Theta] \[Lambda]^2 \[Mu])^2 - 
         4 (a^2 \[Theta] - 2 a k p \[Theta] + k^2 p^2 \[Theta] + 
            b \[Alpha]^2 \[Lambda]^2 + d \[Alpha]^2 \[Lambda]^2 - 
            p \[Alpha]^2 \[Lambda]^2) (\[Alpha]^2 \[Lambda]^2 + \
\[Theta] \[Lambda]^2 - 2 \[Alpha] \[Theta] \[Lambda]^2 + 
            2 \[Theta] \[Lambda]^3 \[Mu] - 
            2 \[Alpha] \[Theta] \[Lambda]^3 \[Mu] + \[Theta] \
\[Lambda]^4 \[Mu]^2)))/(2 (\[Alpha]^2 \[Lambda]^2 + \[Theta] \
\[Lambda]^2 - 2 \[Alpha] \[Theta] \[Lambda]^2 + 
        2 \[Theta] \[Lambda]^3 \[Mu] - 
        2 \[Alpha] \[Theta] \[Lambda]^3 \[Mu] + \[Theta] \[Lambda]^4 \
\[Mu]^2))}, {x -> (
   1/(\[Alpha] \[Lambda]))(-a + k p + (
     a \[Theta] \[Lambda]^2)/(\[Alpha]^2 \[Lambda]^2 + \[Theta] \
\[Lambda]^2 - 2 \[Alpha] \[Theta] \[Lambda]^2 + 
      2 \[Theta] \[Lambda]^3 \[Mu] - 
      2 \[Alpha] \[Theta] \[Lambda]^3 \[Mu] + \[Theta] \[Lambda]^4 \
\[Mu]^2) - (
     k p \[Theta] \[Lambda]^2)/(\[Alpha]^2 \[Lambda]^2 + \[Theta] \
\[Lambda]^2 - 2 \[Alpha] \[Theta] \[Lambda]^2 + 
      2 \[Theta] \[Lambda]^3 \[Mu] - 
      2 \[Alpha] \[Theta] \[Lambda]^3 \[Mu] + \[Theta] \[Lambda]^4 \
\[Mu]^2) - (
     2 a \[Alpha] \[Theta] \[Lambda]^2)/(\[Alpha]^2 \[Lambda]^2 + \
\[Theta] \[Lambda]^2 - 2 \[Alpha] \[Theta] \[Lambda]^2 + 
      2 \[Theta] \[Lambda]^3 \[Mu] - 
      2 \[Alpha] \[Theta] \[Lambda]^3 \[Mu] + \[Theta] \[Lambda]^4 \
\[Mu]^2) + (
     2 k p \[Alpha] \[Theta] \[Lambda]^2)/(\[Alpha]^2 \[Lambda]^2 + \
\[Theta] \[Lambda]^2 - 2 \[Alpha] \[Theta] \[Lambda]^2 + 
      2 \[Theta] \[Lambda]^3 \[Mu] - 
      2 \[Alpha] \[Theta] \[Lambda]^3 \[Mu] + \[Theta] \[Lambda]^4 \
\[Mu]^2) + (
     a \[Alpha]^2 \[Theta] \[Lambda]^2)/(\[Alpha]^2 \[Lambda]^2 + \
\[Theta] \[Lambda]^2 - 2 \[Alpha] \[Theta] \[Lambda]^2 + 
      2 \[Theta] \[Lambda]^3 \[Mu] - 
      2 \[Alpha] \[Theta] \[Lambda]^3 \[Mu] + \[Theta] \[Lambda]^4 \
\[Mu]^2) - (
     k p \[Alpha]^2 \[Theta] \[Lambda]^2)/(\[Alpha]^2 \[Lambda]^2 + \
\[Theta] \[Lambda]^2 - 2 \[Alpha] \[Theta] \[Lambda]^2 + 
      2 \[Theta] \[Lambda]^3 \[Mu] - 
      2 \[Alpha] \[Theta] \[Lambda]^3 \[Mu] + \[Theta] \[Lambda]^4 \
\[Mu]^2) + (
     2 a \[Theta] \[Lambda]^3 \[Mu])/(\[Alpha]^2 \[Lambda]^2 + \
\[Theta] \[Lambda]^2 - 2 \[Alpha] \[Theta] \[Lambda]^2 + 
      2 \[Theta] \[Lambda]^3 \[Mu] - 
      2 \[Alpha] \[Theta] \[Lambda]^3 \[Mu] + \[Theta] \[Lambda]^4 \
\[Mu]^2) - (
     2 k p \[Theta] \[Lambda]^3 \[Mu])/(\[Alpha]^2 \[Lambda]^2 + \
\[Theta] \[Lambda]^2 - 2 \[Alpha] \[Theta] \[Lambda]^2 + 
      2 \[Theta] \[Lambda]^3 \[Mu] - 
      2 \[Alpha] \[Theta] \[Lambda]^3 \[Mu] + \[Theta] \[Lambda]^4 \
\[Mu]^2) - (
     2 a \[Alpha] \[Theta] \[Lambda]^3 \[Mu])/(\[Alpha]^2 \[Lambda]^2 \
+ \[Theta] \[Lambda]^2 - 2 \[Alpha] \[Theta] \[Lambda]^2 + 
      2 \[Theta] \[Lambda]^3 \[Mu] - 
      2 \[Alpha] \[Theta] \[Lambda]^3 \[Mu] + \[Theta] \[Lambda]^4 \
\[Mu]^2) + (
     2 k p \[Alpha] \[Theta] \[Lambda]^3 \[Mu])/(\[Alpha]^2 \
\[Lambda]^2 + \[Theta] \[Lambda]^2 - 
      2 \[Alpha] \[Theta] \[Lambda]^2 + 
      2 \[Theta] \[Lambda]^3 \[Mu] - 
      2 \[Alpha] \[Theta] \[Lambda]^3 \[Mu] + \[Theta] \[Lambda]^4 \
\[Mu]^2) + (
     a \[Theta] \[Lambda]^4 \[Mu]^2)/(\[Alpha]^2 \[Lambda]^2 + \
\[Theta] \[Lambda]^2 - 2 \[Alpha] \[Theta] \[Lambda]^2 + 
      2 \[Theta] \[Lambda]^3 \[Mu] - 
      2 \[Alpha] \[Theta] \[Lambda]^3 \[Mu] + \[Theta] \[Lambda]^4 \
\[Mu]^2) - (
     k p \[Theta] \[Lambda]^4 \[Mu]^2)/(\[Alpha]^2 \[Lambda]^2 + \
\[Theta] \[Lambda]^2 - 2 \[Alpha] \[Theta] \[Lambda]^2 + 
      2 \[Theta] \[Lambda]^3 \[Mu] - 
      2 \[Alpha] \[Theta] \[Lambda]^3 \[Mu] + \[Theta] \[Lambda]^4 \
\[Mu]^2) - (\[Lambda] \[Sqrt]((2 a \[Theta] \[Lambda] - 
             2 k p \[Theta] \[Lambda] - 
             2 a \[Alpha] \[Theta] \[Lambda] + 
             2 k p \[Alpha] \[Theta] \[Lambda] + 
             2 a \[Theta] \[Lambda]^2 \[Mu] - 
             2 k p \[Theta] \[Lambda]^2 \[Mu])^2 - 
           4 (a^2 \[Theta] - 2 a k p \[Theta] + k^2 p^2 \[Theta] + 
              b \[Alpha]^2 \[Lambda]^2 + d \[Alpha]^2 \[Lambda]^2 - 
              p \[Alpha]^2 \[Lambda]^2) (\[Alpha]^2 \[Lambda]^2 + \
\[Theta] \[Lambda]^2 - 2 \[Alpha] \[Theta] \[Lambda]^2 + 
              2 \[Theta] \[Lambda]^3 \[Mu] - 
              2 \[Alpha] \[Theta] \[Lambda]^3 \[Mu] + \[Theta] \
\[Lambda]^4 \[Mu]^2)))/(2 (\[Alpha]^2 \[Lambda]^2 + \[Theta] \
\[Lambda]^2 - 2 \[Alpha] \[Theta] \[Lambda]^2 + 
          2 \[Theta] \[Lambda]^3 \[Mu] - 
          2 \[Alpha] \[Theta] \[Lambda]^3 \[Mu] + \[Theta] \
\[Lambda]^4 \[Mu]^2)) + (\[Alpha] \[Lambda] \[Sqrt]((2 a \[Theta] \
\[Lambda] - 2 k p \[Theta] \[Lambda] - 
             2 a \[Alpha] \[Theta] \[Lambda] + 
             2 k p \[Alpha] \[Theta] \[Lambda] + 
             2 a \[Theta] \[Lambda]^2 \[Mu] - 
             2 k p \[Theta] \[Lambda]^2 \[Mu])^2 - 
           4 (a^2 \[Theta] - 2 a k p \[Theta] + k^2 p^2 \[Theta] + 
              b \[Alpha]^2 \[Lambda]^2 + d \[Alpha]^2 \[Lambda]^2 - 
              p \[Alpha]^2 \[Lambda]^2) (\[Alpha]^2 \[Lambda]^2 + \
\[Theta] \[Lambda]^2 - 2 \[Alpha] \[Theta] \[Lambda]^2 + 
              2 \[Theta] \[Lambda]^3 \[Mu] - 
              2 \[Alpha] \[Theta] \[Lambda]^3 \[Mu] + \[Theta] \
\[Lambda]^4 \[Mu]^2)))/(2 (\[Alpha]^2 \[Lambda]^2 + \[Theta] \
\[Lambda]^2 - 2 \[Alpha] \[Theta] \[Lambda]^2 + 
          2 \[Theta] \[Lambda]^3 \[Mu] - 
          2 \[Alpha] \[Theta] \[Lambda]^3 \[Mu] + \[Theta] \
\[Lambda]^4 \[Mu]^2)) - (\[Lambda]^2 \[Mu] \[Sqrt]((2 a \[Theta] \
\[Lambda] - 2 k p \[Theta] \[Lambda] - 
             2 a \[Alpha] \[Theta] \[Lambda] + 
             2 k p \[Alpha] \[Theta] \[Lambda] + 
             2 a \[Theta] \[Lambda]^2 \[Mu] - 
             2 k p \[Theta] \[Lambda]^2 \[Mu])^2 - 
           4 (a^2 \[Theta] - 2 a k p \[Theta] + k^2 p^2 \[Theta] + 
              b \[Alpha]^2 \[Lambda]^2 + d \[Alpha]^2 \[Lambda]^2 - 
              p \[Alpha]^2 \[Lambda]^2) (\[Alpha]^2 \[Lambda]^2 + \
\[Theta] \[Lambda]^2 - 2 \[Alpha] \[Theta] \[Lambda]^2 + 
              2 \[Theta] \[Lambda]^3 \[Mu] - 
              2 \[Alpha] \[Theta] \[Lambda]^3 \[Mu] + \[Theta] \
\[Lambda]^4 \[Mu]^2)))/(2 (\[Alpha]^2 \[Lambda]^2 + \[Theta] \
\[Lambda]^2 - 2 \[Alpha] \[Theta] \[Lambda]^2 + 
          2 \[Theta] \[Lambda]^3 \[Mu] - 
          2 \[Alpha] \[Theta] \[Lambda]^3 \[Mu] + \[Theta] \
\[Lambda]^4 \[Mu]^2))), 
  y -> (-2 a \[Theta] \[Lambda] + 2 k p \[Theta] \[Lambda] + 
      2 a \[Alpha] \[Theta] \[Lambda] - 
      2 k p \[Alpha] \[Theta] \[Lambda] - 
      2 a \[Theta] \[Lambda]^2 \[Mu] + 
      2 k p \[Theta] \[Lambda]^2 \[Mu] + \[Sqrt]((2 a \[Theta] \
\[Lambda] - 2 k p \[Theta] \[Lambda] - 
           2 a \[Alpha] \[Theta] \[Lambda] + 
           2 k p \[Alpha] \[Theta] \[Lambda] + 
           2 a \[Theta] \[Lambda]^2 \[Mu] - 
           2 k p \[Theta] \[Lambda]^2 \[Mu])^2 - 
         4 (a^2 \[Theta] - 2 a k p \[Theta] + k^2 p^2 \[Theta] + 
            b \[Alpha]^2 \[Lambda]^2 + d \[Alpha]^2 \[Lambda]^2 - 
            p \[Alpha]^2 \[Lambda]^2) (\[Alpha]^2 \[Lambda]^2 + \
\[Theta] \[Lambda]^2 - 2 \[Alpha] \[Theta] \[Lambda]^2 + 
            2 \[Theta] \[Lambda]^3 \[Mu] - 
            2 \[Alpha] \[Theta] \[Lambda]^3 \[Mu] + \[Theta] \
\[Lambda]^4 \[Mu]^2)))/(2 (\[Alpha]^2 \[Lambda]^2 + \[Theta] \
\[Lambda]^2 - 2 \[Alpha] \[Theta] \[Lambda]^2 + 
        2 \[Theta] \[Lambda]^3 \[Mu] - 
        2 \[Alpha] \[Theta] \[Lambda]^3 \[Mu] + \[Theta] \[Lambda]^4 \
\[Mu]^2))}}
7

Thanks for your help. Trying to reduce the variables numbers, but can not reduce all…but know the reason why it can not run. Thanks.

8

Thanks for your help. I tried this method, but still no result, could you guide me your code? Thanks a lot.

9

Thanks for your help. I tried this method, but still no result, could you guide me your .nb code? Thanks a lot.

10

Maybe you can start MMA with a fresh kernel.I’m using Mathematica 10.2 and works fine.

11

Really appreciate your help. My MMA version is 10.2 too, but it is Mac version, not Windows version.
I reduced the symbolic coefficients as much as possible, and still not work, should I transfer Windows version also?

12

Mac or Windows, both versions are the same.MMA has difficulty with this equations.After several hours of computation I’m not saw any result.I do not know why it has such problems.
Only numericall is a quick solution with MMA.
Maple 2015 solves these equations very quickly.

13

Try with this ?

eq1 = FullSimplify[1/2 ? y^(1/2) x^(-(1/2))(p - 1/2 x^2 - 1/2 y^2) + (a - k p + ? (x y)^(1/2)) (-x) == 0]
eq2 = FullSimplify[1/2 ? x^(1/2) y^(-(1/2))(p - 1/2 x^2 - 1/2 y^2) + (a - k p + ? (x y)^(1/2)) (-y) == 0]
Solve[Join[eq1, eq2], {x, y}]

I am waiting and nothing.

Maples solutions after few seconds.

Maplessolutions.png
Maplessolutions.png1096×526 16.5 KB

14

In[1]:= eq1 =
FullSimplify[([Lambda] [Alpha] + [Lambda] [Mu] y) (p - [Theta]
x^2 - (1 - [Theta]) y^2 - b - d) + (a -
k p + [Lambda] ([Alpha] x + (1 - [Alpha]) y + [Lambda]
[Mu] y)) (-2 [Theta] x) == 0];
eq2 = FullSimplify[([Lambda] (1 - [Alpha]) + [Lambda] [Mu] x) (p
- [Theta] x^2 - (1 - [Theta]) y^2 - b - d) + (a -
k p + [Lambda] ([Alpha] x + (1 - [Alpha]) y + [Lambda]
[Mu] y)) (-2 (1 - [Theta]) y) == 0];

In[3]:= AbsoluteTiming[Solve[Join[eq1, eq2], {x, y}];]

Out[3]= {0.627934, Null}

Are you running out of memory? My PC has 16 GBytes of RAM.

15

Yes, you are right, transferring Windows version is not helpful. Let me try Maples. Thanks a lot.

16

Not sure, looks like it is, it just say “calculating…”, and waited several hours, still no result. Thanks.

17

Mr. Frank Kampas you made ??a mistake.

We want to solve these equations:

eq1 = FullSimplify[1/2 ? y^(1/2) x^(-(1/2))(p - 1/2 x^2 - 1/2 y^2) + (a - k p + ? (x y)^(1/2)) (-x) == 0]
eq2 = FullSimplify[1/2 ? x^(1/2) y^(-(1/2))(p - 1/2 x^2 - 1/2 y^2) + (a - k p + ? (x y)^(1/2)) (-y) == 0]

not those:

eq1 = FullSimplify[(? ? + ? ? y) (p - ? x^2 - (1 - ?) y^2 - b - d) + (a - k p + ? (? x + (1 - ?) y + ? ? y)) (-2 ? x) == 0]
eq2 = FullSimplify[(? (1 - ?) + ? ? x) (p - ? x^2 - (1 - ?) y^2 - b - d) + (a - k p + ? (? x + (1 - ?) y + ? ? y)) (-2 (1 - ?) y) == 0]

My Laptop has 4 GBytes of RAM.With these 2 equations Kernel level of consumption of RAM is only 55MB and did not increase for the next hours calculating.

See file attachments:

18

With a bit of cheating we can get candidate solutions for x^2 and y^2 as below. The caveat is that they might be incorrect due to branch cut issues.

 exprs = {1/2 \[Lambda] y^(1/2)
      x^(-(1/2)) (p - 1/2 x^2 - 1/2 y^2) + (a - 
       k p + \[Lambda] (x y)^(1/2)) (-x), 
   1/2 \[Lambda] x^(1/2)
      y^(-(1/2)) (p - 1/2 x^2 - 1/2 y^2) + (a - 
       k p + \[Lambda] (x y)^(1/2)) (-y)};
e2 = Numerator[Together[exprs]];
e3 = PowerExpand[e2 /. {x -> x^2, y -> y^2}]

(* Out[11]= {-4 a x^3 + 4 k p x^3 + 2 p y \[Lambda] - 5 x^4 y \[Lambda] -
   y^5 \[Lambda], -4 a y^3 + 4 k p y^3 + 2 p x \[Lambda] - 
  x^5 \[Lambda] - 5 x y^4 \[Lambda]} *)

Solving this modified system is not hard.

Timing[solnsb = Solve[e3 == 0, {x, y}];]

Out[21]= {1.356464, Null}

There are 25 solutions. I show the first three are shown below. One would need to take square roots though, since we solved (modulo branch cut issues) for (x^2, y^2) pairs rather than (x,y).

{{x -> 0, 
  y -> 0}, {x -> (
   1/(-2 a p \[Lambda] + 
    2 k p^2 \[Lambda]))((
     4 a^2 (3 p - (2 a^2)/\[Lambda]^2 + (4 a k p)/\[Lambda]^2 - (
        2 k^2 p^2)/\[Lambda]^2 - (
        2 Sqrt[(a - k p)^2 (a^2 - 2 a k p + k^2 p^2 - 
            3 p \[Lambda]^2)])/\[Lambda]^2)^(3/4))/(3 Sqrt[3]) - (
     8 a k p (3 p - (2 a^2)/\[Lambda]^2 + (4 a k p)/\[Lambda]^2 - (
        2 k^2 p^2)/\[Lambda]^2 - (
        2 Sqrt[(a - k p)^2 (a^2 - 2 a k p + k^2 p^2 - 
            3 p \[Lambda]^2)])/\[Lambda]^2)^(3/4))/(3 Sqrt[3]) + (
     4 k^2 p^2 (3 p - (2 a^2)/\[Lambda]^2 + (4 a k p)/\[Lambda]^2 - (
        2 k^2 p^2)/\[Lambda]^2 - (
        2 Sqrt[(a - k p)^2 (a^2 - 2 a k p + k^2 p^2 - 
            3 p \[Lambda]^2)])/\[Lambda]^2)^(3/4))/(3 Sqrt[3]) - (
     p \[Lambda]^2 (3 p - (2 a^2)/\[Lambda]^2 + (
        4 a k p)/\[Lambda]^2 - (2 k^2 p^2)/\[Lambda]^2 - (
        2 Sqrt[(a - k p)^2 (a^2 - 2 a k p + k^2 p^2 - 
            3 p \[Lambda]^2)])/\[Lambda]^2)^(3/4))/Sqrt[
     3] + (\[Lambda]^2 (3 p - (2 a^2)/\[Lambda]^2 + (
        4 a k p)/\[Lambda]^2 - (2 k^2 p^2)/\[Lambda]^2 - ({{x -> 0, 
          y -> 0}, {x -> (
           1/(-2 a p \[Lambda] + 
            2 k p^2 \[Lambda]))((
             4 a^2 (3 p - (2 a^2)/\[Lambda]^2 + (4 a k p)/\[Lambda]^2 - (
                2 k^2 p^2)/\[Lambda]^2 - (
                2 Sqrt[(a - k p)^2 (a^2 - 2 a k p + k^2 p^2 - 
                    3 p \[Lambda]^2)])/\[Lambda]^2)^(3/4))/(3 Sqrt[3]) - (
             8 a k p (3 p - (2 a^2)/\[Lambda]^2 + (4 a k p)/\[Lambda]^2 - (
                2 k^2 p^2)/\[Lambda]^2 - (
                2 Sqrt[(a - k p)^2 (a^2 - 2 a k p + k^2 p^2 - 
                    3 p \[Lambda]^2)])/\[Lambda]^2)^(3/4))/(3 Sqrt[3]) + (
             4 k^2 p^2 (3 p - (2 a^2)/\[Lambda]^2 + (4 a k p)/\[Lambda]^2 - (
                2 k^2 p^2)/\[Lambda]^2 - (
                2 Sqrt[(a - k p)^2 (a^2 - 2 a k p + k^2 p^2 - 
                    3 p \[Lambda]^2)])/\[Lambda]^2)^(3/4))/(3 Sqrt[3]) - (
             p \[Lambda]^2 (3 p - (2 a^2)/\[Lambda]^2 + (
                4 a k p)/\[Lambda]^2 - (2 k^2 p^2)/\[Lambda]^2 - (
                2 Sqrt[(a - k p)^2 (a^2 - 2 a k p + k^2 p^2 - 
                    3 p \[Lambda]^2)])/\[Lambda]^2)^(3/4))/Sqrt[
             3] + (\[Lambda]^2 (3 p - (2 a^2)/\[Lambda]^2 + (
                4 a k p)/\[Lambda]^2 - (2 k^2 p^2)/\[Lambda]^2 - (
                2 Sqrt[(a - k p)^2 (a^2 - 2 a k p + k^2 p^2 - 
                    3 p \[Lambda]^2)])/\[Lambda]^2)^(7/4))/(3 Sqrt[3])), 
          y -> -((3 p - (2 a^2)/\[Lambda]^2 + (4 a k p)/\[Lambda]^2 - (
              2 k^2 p^2)/\[Lambda]^2 - (
              2 Sqrt[(a - k p)^2 (a^2 - 2 a k p + k^2 p^2 - 
                  3 p \[Lambda]^2)])/\[Lambda]^2)^(1/4)/Sqrt[3])},
{x -> (1/(-2 a p \[Lambda] + 
   2 k p^2 \[Lambda]))(-((
     4 I a^2 (3 p - (2 a^2)/\[Lambda]^2 + (4 a k p)/\[Lambda]^2 - (
        2 k^2 p^2)/\[Lambda]^2 - (
        2 Sqrt[(a - k p)^2 (a^2 - 2 a k p + k^2 p^2 - 
            3 p \[Lambda]^2)])/\[Lambda]^2)^(3/4))/(3 Sqrt[3])) + (
    8 I a k p (3 p - (2 a^2)/\[Lambda]^2 + (4 a k p)/\[Lambda]^2 - (
       2 k^2 p^2)/\[Lambda]^2 - (
       2 Sqrt[(a - k p)^2 (a^2 - 2 a k p + k^2 p^2 - 
           3 p \[Lambda]^2)])/\[Lambda]^2)^(3/4))/(3 Sqrt[3]) - (
    4 I k^2 p^2 (3 p - (2 a^2)/\[Lambda]^2 + (4 a k p)/\[Lambda]^2 - (
       2 k^2 p^2)/\[Lambda]^2 - (
       2 Sqrt[(a - k p)^2 (a^2 - 2 a k p + k^2 p^2 - 
           3 p \[Lambda]^2)])/\[Lambda]^2)^(3/4))/(3 Sqrt[3]) + (
    I p \[Lambda]^2 (3 p - (2 a^2)/\[Lambda]^2 + (
       4 a k p)/\[Lambda]^2 - (2 k^2 p^2)/\[Lambda]^2 - (
       2 Sqrt[(a - k p)^2 (a^2 - 2 a k p + k^2 p^2 - 
           3 p \[Lambda]^2)])/\[Lambda]^2)^(3/4))/Sqrt[3] - (
    I \[Lambda]^2 (3 p - (2 a^2)/\[Lambda]^2 + (
       4 a k p)/\[Lambda]^2 - (2 k^2 p^2)/\[Lambda]^2 - (
       2 Sqrt[(a - k p)^2 (a^2 - 2 a k p + k^2 p^2 - 
           3 p \[Lambda]^2)])/\[Lambda]^2)^(7/4))/(3 Sqrt[3])), 
 y -> -((I (3 p - (2 a^2)/\[Lambda]^2 + (4 a k p)/\[Lambda]^2 - (
      2 k^2 p^2)/\[Lambda]^2 - (
      2 Sqrt[(a - k p)^2 (a^2 - 2 a k p + k^2 p^2 - 
          3 p \[Lambda]^2)])/\[Lambda]^2)^(1/4))/Sqrt[3])}
..
19

Cheating or not MMA has problems with this equation.
I hope that development team fix this issue, will improve in a future release.

Thanks for Reply.