closed form solution in a system of equation

Dear friends, Hi

I am trying to find closed form solution for a system of equations (8 equations and 8 variables). All the equations are both non-linear and parametric.

I used "solve" in coding but I faced this massage: no explicit solution.

How could I improve my codes for having the parametric solution? Is there any other function?

[Mahsa Babaee]

syms X1 X2 X3 X4 X5 X6 X7 X8 X9;

syms h k1 Lz1 k21 Lz2 k32 Lz3 a1 a2 a3 b1 b2 b3 w1 w2 w3 r1 r2 r3;

eq1 = X1-1 == 0;

eq2 = h*X1-k1*X7*X4==0;

eq3 = X1*cos(X7*Lz1)+X4*sin(X7*Lz1)-X2==0;

eq4 = -X1*X6*sin(X7*Lz1)+X4*X6*cos(X7*Lz1)-X8*X5*k21==0;

eq5 = X2*cos(X8*Lz2)+X5*sin(X8*Lz2)-X3==0;

eq6 = -X2*X8*sin(X8*Lz2)+X5*X8*cos(X8*Lz2)-X9*X6*k32==0;

eq7 = X3*cos(X9*Lz3)+X6*sin(X9*Lz3)==0;

eq8 = (a1/a2)*(b1^2+r1^2+X7^2+w1^2)-b2^2-r2^2-w2^2-X8^2 ==0;

eq9 = (a1/a3)*(b1^2+r1^2+X7^2+w1^2)-b3^2-r3^2-w3^2-X9^2 ==0;

sol = solve([eq1,eq2,eq3,eq4,eq5,eq6,eq7,eq8,eq9],[X1 X2 X3 X4 X5 X6 X7 X8 X9]);

Answer 1

[Walter Roberson]

There are four families of solutions.

[X1 == 1,
X2 == (cos(Lz1*h*sin(RootOf(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*h^2))/cos(RootOf(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*h^2))/k1)*sin(RootOf(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*h^2))+cos(RootOf(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*h^2))*sin(Lz1*h*sin(RootOf(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*h^2))/cos(RootOf(Lz1^2*cos(Z)^2*h^2+cos(Z)^
2*Z^2*k1^2-Lz1^2*h^2))/k1))/sin(RootOf(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*h^2)),
X3 == cos(Lz2*RootOf((Lz1^2*a2*h^2+RootOf(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*h^2)^2*a2*k1^2)*Z^2-Lz1^2*a1*b1^2*h^2*k1^2-Lz1^2*a1*h^2*k1^2*r1^2-Lz1^2*a1*h^2*k1^2*w1^2+Lz1^2*a2*b2^2*h^2*k1^2+Lz1^2*a2*h^2*k1^2*r2^2+Lz1^2*a2*h^2*k1^2*w2^2-RootOf(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*h^2)^2*a1*h^2*k1^2)/cos(RootOf(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*h^2))/k1)*(cos(Lz1*h*sin(RootOf(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*h^2))/cos(RootOf(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*h^2))/k1)*sin(RootOf(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*h^2))+cos(RootOf(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*h^2))*sin(Lz1*h*sin(RootOf(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*h^2))/cos(RootOf(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*h^2))/k1))/sin(RootOf(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*h^2)),
X4 == cos(RootOf(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*h^2))/sin(RootOf(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*h^2)),
X5 == 0,
X6 == 0,
X7 == h*sin(RootOf(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*h^2))/cos(RootOf(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*h^2))/k1,
X8 == RootOf((Lz1^2*a2*h^2+RootOf(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*h^2)^2*a2*k1^2)*Z^2-Lz1^2*a1*b1^2*h^2*k1^2-Lz1^2*a1*h^2*k1^2*r1^2-Lz1^2*a1*h^2*k1^2*w1^2+Lz1^2*a2*b2^2*h^2*k1^2+Lz1^2*a2*h^2*k1^2*r2^2+Lz1^2*a2*h^2*k1^2*w2^2-RootOf(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*h^2)^2*a1*h^2*k1^2)/cos(RootOf(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*h^2))/k1,
X9 == RootOf((Lz1^2*a3*h^2+RootOf(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*h^2)^2*a3*k1^2)*Z^2-Lz1^2*a1*b1^2*h^2*k1^2-Lz1^2*a1*h^2*k1^2*r1^2-Lz1^2*a1*h^2*k1^2*w1^2+Lz1^2*a3*b3^2*h^2*k1^2+Lz1^2*a3*h^2*k1^2*r3^2+Lz1^2*a3*h^2*k1^2*w3^2-RootOf(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*h^2)^2*a1*h^2*k1^2)/cos(RootOf(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*h^2))/k1]

[X1 == 1,
X2 == (cos(Lz1*h*sin(RootOf(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*k1^2-Lz1
^2*h^2))/cos(RootOf(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*h^2))/k1)*sin
(RootOf(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*h^2))+cos(RootOf(Lz1^2*
cos(Z)^2*h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*h^2))*sin(Lz1*h*sin(RootOf(Lz1^2*cos(Z
)^2*h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*h^2))/cos(RootOf(Lz1^2*cos(Z)^2*h^2+cos(Z)^
2*Z^2*k1^2-Lz1^2*h^2))/k1))/sin(RootOf(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*k1^2
-Lz1^2*h^2)),
X3 == cos(Lz2*RootOf((Lz1^2*a2*h^2+RootOf(Lz1^2*cos(Z)^2*h^2+cos(
Z)^2*Z^2*k1^2-Lz1^2*h^2)^2*a2*k1^2)*Z^2-Lz1^2*a1*b1^2*h^2*k1^2-Lz1^2*a1*h^2*
k1^2*r1^2-Lz1^2*a1*h^2*k1^2*w1^2+Lz1^2*a2*b2^2*h^2*k1^2+Lz1^2*a2*h^2*k1^2*r2^2+
Lz1^2*a2*h^2*k1^2*w2^2-RootOf(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*h^2
)^2*a1*h^2*k1^2)/cos(RootOf(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*h^2))
/k1)*(cos(Lz1*h*sin(RootOf(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*h^2))/
cos(RootOf(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*h^2))/k1)*sin(RootOf(
Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*h^2))+cos(RootOf(Lz1^2*cos(Z)^2*
h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*h^2))*sin(Lz1*h*sin(RootOf(Lz1^2*cos(Z)^2*h^2+
cos(Z)^2*Z^2*k1^2-Lz1^2*h^2))/cos(RootOf(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*
k1^2-Lz1^2*h^2))/k1))/sin(RootOf(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*
h^2)),
X4 == -cos(RootOf(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*h^2))/sin
(RootOf(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*h^2)),
X5 == 0,
X6 == 0,
X7
== -h*sin(RootOf(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*h^2))/cos(RootOf(
Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*h^2))/k1,
X8 == RootOf((Lz1^2*a2*h
^2+RootOf(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*h^2)^2*a2*k1^2)*Z^2-
Lz1^2*a1*b1^2*h^2*k1^2-Lz1^2*a1*h^2*k1^2*r1^2-Lz1^2*a1*h^2*k1^2*w1^2+Lz1^2*a2*
b2^2*h^2*k1^2+Lz1^2*a2*h^2*k1^2*r2^2+Lz1^2*a2*h^2*k1^2*w2^2-RootOf(Lz1^2*cos(Z
)^2*h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*h^2)^2*a1*h^2*k1^2)/cos(RootOf(Lz1^2*cos(Z)^
2*h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*h^2))/k1,
X9 == RootOf((Lz1^2*a3*h^2+RootOf(Lz1^
2*cos(Z)^2*h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*h^2)^2*a3*k1^2)*Z^2-Lz1^2*a1*b1^2*h^
2*k1^2-Lz1^2*a1*h^2*k1^2*r1^2-Lz1^2*a1*h^2*k1^2*w1^2+Lz1^2*a3*b3^2*h^2*k1^2+Lz1
^2*a3*h^2*k1^2*r3^2+Lz1^2*a3*h^2*k1^2*w3^2-RootOf(Lz1^2*cos(Z)^2*h^2+cos(Z)^2
*Z^2*k1^2-Lz1^2*h^2)^2*a1*h^2*k1^2)/cos(RootOf(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*
Z^2*k1^2-Lz1^2*h^2))/k1]

[X1 == 1,
X2 == (cos(Lz1*h*sin(RootOf(cos(Z)^2*(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z
^2*k1^2-Lz1^2*h^2)))/cos(RootOf(cos(Z)^2*(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*
k1^2-Lz1^2*h^2)))/k1)*sin(RootOf(cos(Z)^2*(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*
k1^2-Lz1^2*h^2)))+cos(RootOf(cos(Z)^2*(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*k1^2
-Lz1^2*h^2)))*sin(Lz1*h*sin(RootOf(cos(Z)^2*(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^
2*k1^2-Lz1^2*h^2)))/cos(RootOf(cos(Z)^2*(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*k1
^2-Lz1^2*h^2)))/k1))/sin(RootOf(cos(Z)^2*(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*
k1^2-Lz1^2*h^2))),
X3 == cos(Lz2/k1*RootOf(-a1*b1^2*k1^2*cos(RootOf(Lz1*sin(Z)*
h-Z*k1*cos(Z)))^2-a1*k1^2*r1^2*cos(RootOf(Lz1*sin(Z)*h-Z*k1*cos(Z)))^2-a1*
k1^2*w1^2*cos(RootOf(Lz1*sin(Z)*h-Z*k1*cos(Z)))^2+a2*b2^2*k1^2*cos(RootOf(
Lz1*sin(Z)*h-Z*k1*cos(Z)))^2+a2*k1^2*r2^2*cos(RootOf(Lz1*sin(Z)*h-Z*k1*cos
(Z)))^2+a2*k1^2*w2^2*cos(RootOf(Lz1*sin(Z)*h-Z*k1*cos(Z)))^2+Z^2*a2*cos(
RootOf(Lz1*sin(Z)*h-Z*k1*cos(Z)))^2+cos(RootOf(Lz1*sin(Z)*h-Z*k1*cos(Z)))
^2*a1*h^2-h^2*a1))*(cos(Lz1*h*sin(RootOf(cos(Z)^2*(Lz1^2*cos(Z)^2*h^2+cos(Z)
^2*Z^2*k1^2-Lz1^2*h^2)))/cos(RootOf(cos(Z)^2*(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*
Z^2*k1^2-Lz1^2*h^2)))/k1)*sin(RootOf(cos(Z)^2*(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*
Z^2*k1^2-Lz1^2*h^2)))+cos(RootOf(cos(Z)^2*(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2
*k1^2-Lz1^2*h^2)))*sin(Lz1*h*sin(RootOf(cos(Z)^2*(Lz1^2*cos(Z)^2*h^2+cos(Z)^
2*Z^2*k1^2-Lz1^2*h^2)))/cos(RootOf(cos(Z)^2*(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z
^2*k1^2-Lz1^2*h^2)))/k1))/sin(RootOf(cos(Z)^2*(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*
Z^2*k1^2-Lz1^2*h^2))),
X4 == -cos(RootOf(cos(Z)^2*(Lz1^2*cos(Z)^2*h^2+cos(Z)
^2*Z^2*k1^2-Lz1^2*h^2)))/sin(RootOf(cos(Z)^2*(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*
Z^2*k1^2-Lz1^2*h^2))),
X5 == 0,
X6 == 0,
X7 == -h*sin(RootOf(cos(Z)^2*(Lz1^2*cos
(Z)^2*h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*h^2)))/cos(RootOf(cos(Z)^2*(Lz1^2*cos(Z)
^2*h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*h^2)))/k1,
X8 == RootOf(-a1*b1^2*k1^2-tan(
RootOf(Lz1*tan(Z)*h-Z*k1))^2*a1*h^2-a1*k1^2*r1^2-a1*k1^2*w1^2+a2*b2^2*k1^2+a2
*k1^2*r2^2+a2*k1^2*w2^2+Z^2*a2)/k1,
X9 == RootOf(-a1*b1^2*k1^2-tan(RootOf(Lz1*
tan(Z)*h-Z*k1))^2*a1*h^2-a1*k1^2*r1^2-a1*k1^2*w1^2+a3*b3^2*k1^2+a3*k1^2*r3^2+
a3*k1^2*w3^2+Z^2*a3)/k1]

[X1 == 1,
X2 == (cos(Lz1*h*sin(RootOf(cos(Z)^2*(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z
^2*k1^2-Lz1^2*h^2)))/cos(RootOf(cos(Z)^2*(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*
k1^2-Lz1^2*h^2)))/k1)*sin(RootOf(cos(Z)^2*(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*
k1^2-Lz1^2*h^2)))+cos(RootOf(cos(Z)^2*(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*k1^2
-Lz1^2*h^2)))*sin(Lz1*h*sin(RootOf(cos(Z)^2*(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^
2*k1^2-Lz1^2*h^2)))/cos(RootOf(cos(Z)^2*(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*k1
^2-Lz1^2*h^2)))/k1))/sin(RootOf(cos(Z)^2*(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2*
k1^2-Lz1^2*h^2))),
X3 == cos(Lz2/k1*RootOf(-a1*b1^2*k1^2*cos(RootOf(Lz1*sin(Z)*
h-Z*k1*cos(Z)))^2-a1*k1^2*r1^2*cos(RootOf(Lz1*sin(Z)*h-Z*k1*cos(Z)))^2-a1*
k1^2*w1^2*cos(RootOf(Lz1*sin(Z)*h-Z*k1*cos(Z)))^2+a2*b2^2*k1^2*cos(RootOf(
Lz1*sin(Z)*h-Z*k1*cos(Z)))^2+a2*k1^2*r2^2*cos(RootOf(Lz1*sin(Z)*h-Z*k1*cos
(Z)))^2+a2*k1^2*w2^2*cos(RootOf(Lz1*sin(Z)*h-Z*k1*cos(Z)))^2+Z^2*a2*cos(
RootOf(Lz1*sin(Z)*h-Z*k1*cos(Z)))^2+cos(RootOf(Lz1*sin(Z)*h-Z*k1*cos(Z)))
^2*a1*h^2-h^2*a1))*(cos(Lz1*h*sin(RootOf(cos(Z)^2*(Lz1^2*cos(Z)^2*h^2+cos(Z)
^2*Z^2*k1^2-Lz1^2*h^2)))/cos(RootOf(cos(Z)^2*(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*
Z^2*k1^2-Lz1^2*h^2)))/k1)*sin(RootOf(cos(Z)^2*(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*
Z^2*k1^2-Lz1^2*h^2)))+cos(RootOf(cos(Z)^2*(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z^2
*k1^2-Lz1^2*h^2)))*sin(Lz1*h*sin(RootOf(cos(Z)^2*(Lz1^2*cos(Z)^2*h^2+cos(Z)^
2*Z^2*k1^2-Lz1^2*h^2)))/cos(RootOf(cos(Z)^2*(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z
^2*k1^2-Lz1^2*h^2)))/k1))/sin(RootOf(cos(Z)^2*(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*
Z^2*k1^2-Lz1^2*h^2))),
X4 == cos(RootOf(cos(Z)^2*(Lz1^2*cos(Z)^2*h^2+cos(Z)^
2*Z^2*k1^2-Lz1^2*h^2)))/sin(RootOf(cos(Z)^2*(Lz1^2*cos(Z)^2*h^2+cos(Z)^2*Z
^2*k1^2-Lz1^2*h^2))),
X5 == 0,
X6 == 0,
X7 == h*sin(RootOf(cos(Z)^2*(Lz1^2*cos(Z
)^2*h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*h^2)))/cos(RootOf(cos(Z)^2*(Lz1^2*cos(Z)^2*
h^2+cos(Z)^2*Z^2*k1^2-Lz1^2*h^2)))/k1,
X8 == RootOf(-a1*b1^2*k1^2-tan(RootOf(
Lz1*tan(Z)*h-Z*k1))^2*a1*h^2-a1*k1^2*r1^2-a1*k1^2*w1^2+a2*b2^2*k1^2+a2*k1^2*
r2^2+a2*k1^2*w2^2+Z^2*a2)/k1,
X9 == RootOf(-a1*b1^2*k1^2-tan(RootOf(Lz1*tan(Z)
*h-Z*k1))^2*a1*h^2-a1*k1^2*r1^2-a1*k1^2*w1^2+a3*b3^2*k1^2+a3*k1^2*r3^2+a3*k1^2
*w3^2+Z^2*a3)/k1]

In the above, RootOf(expression,Z) stands for the set of values of Z such that the expression becomes zero -- the roots of the equations.

[Walter Roberson]

See also https://www.mathworks.com/matlabcentral/answers/756814-solve-non-linear-equations-system-parametrically-why-are-the-answers-empty#comment_1360229