# solution for symbolic equation: matlab can't find a solution for my equation

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Antonio Benigni on 27 Nov 2021
Commented: Walter Roberson on 30 Nov 2021
i have this equation
J(x) =(4*C*Mo*mom*mom1*mus*sin(o)*(C^2 + 3*x^2))/(l*pi*(C^2 - x^2)^3) - (12*C*Mo*mom*mom1*x*cos(o)*(C^2 - x^2)^3*(C^2 + x^2))/(pi*(C + x)^7*(C - x)^7)==0
x is the only variable and the other parameters are positive real numbers
i try to find a solution for x based on the others parameter (there are many solutions, i found it on paper), but using the command solve i get this
root(3*mus*z^4*sin(o) + 3*l*z^3*cos(o) - 2*C^2*mus*z^2*sin(o) + 3*C^2*l*z*cos(o) - C^4*mus*sin(o), z)
this is the same solution i found on paper and thats how i know there are at least 2 real solutions but the program doesn't go on and don't give me the actual family of solution
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### Answers (1)

Paul on 27 Nov 2021
Use the ReturnConditions option to see all of the solutions
syms x C Mo mom mom1 mus o l
J(x) =(4*C*Mo*mom*mom1*mus*sin(o)*(C^2 + 3*x^2))/(l*sym(pi)*(C^2 - x^2)^3) - (12*C*Mo*mom*mom1*x*cos(o)*(C^2 - x^2)^3*(C^2 + x^2))/(sym(pi)*(C + x)^7*(C - x)^7)==0;
sol = solve(J(x),x,'ReturnConditions',true)
sol = struct with fields:
x: [4×1 sym] parameters: [1×0 sym] conditions: [4×1 sym]
sol.x
ans =
However, specifying all the parameters as real and positive yields a much different result
syms x C Mo mom mom1 mus o l positive
J(x) =(4*C*Mo*mom*mom1*mus*sin(o)*(C^2 + 3*x^2))/(l*sym(pi)*(C^2 - x^2)^3) - (12*C*Mo*mom*mom1*x*cos(o)*(C^2 - x^2)^3*(C^2 + x^2))/(sym(pi)*(C + x)^7*(C - x)^7)==0;
sol = solve(J(x),x,'ReturnConditions',true)
sol = struct with fields:
x: z1 parameters: z1 conditions: C ~= z1 & mus*sin(o)*C^4 + 2*mus*sin(o)*C^2*z1^2 == 3*l*cos(o)*C^2*z1 + 3*mus*sin(o)*z1^4 + 3*l*cos(o)*z1^3 & ~in(o/pi, 'integer') & 0 < z1
Ignoring analytic constraints leads to something a bit simpler
syms x C Mo mom mom1 mus o l positive
J(x) =(4*C*Mo*mom*mom1*mus*sin(o)*(C^2 + 3*x^2))/(l*sym(pi)*(C^2 - x^2)^3) - (12*C*Mo*mom*mom1*x*cos(o)*(C^2 - x^2)^3*(C^2 + x^2))/(sym(pi)*(C + x)^7*(C - x)^7)==0;
sol = solve(J(x),x,'ReturnConditions',true,'IgnoreAnalyticConstraints',true)
sol = struct with fields:
x: z1 parameters: z1 conditions: 0 < z1 & mus*sin(o)*C^4 + 2*mus*sin(o)*C^2*z1^2 - 3*l*cos(o)*C^2*z1 - 3*mus*sin(o)*z1^4 - 3*l*cos(o)*z1^3 == 0
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Walter Roberson on 30 Nov 2021
Use 'Maxdegree', 4

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