Solve system of equations when plotting

Hi,

I'm trying to solve a system of equations below symbolically. The challenge is that it only works symbolically if variables 'phi' and 'psi' are defined with numeric values as opposed to symbolic variables (this makes sense, since when 'phi' and 'psi' are defined, the system becomes a system of 5 equations and 5 unknowns as opposed to 5 equations and 7 unknowns.

This works:

t = 1;
t2 = 1;
k = 1;
k2 = 1;
a1 = 1;
phi = pi/3;
psi = pi/6;
syms a2 a3 b1 b2 b3 
S = solve(b1 == a1*t + a2*1j*k,b2 == a1*1j*k + a2*t, b3==a3*t2 + b2*1j*k2*exp(1j*phi), ...
    a2 == exp(1j*phi)*(b2*t2*exp(1j*phi)+a3*1j*k2), a3 == b3*exp(2j*psi));
S.b1

ans = 

This does not

t = 1;
t2 = 1;
k = 1;
k2 = 1;
a1 = 1;
syms a2 a3 b1 b2 b3 phi psi
S = solve(b1 == a1*t + a2*1j*k,b2 == a1*1j*k + a2*t, b3==a3*t2 + b2*1j*k2*exp(1j*phi), ...
    a2 == exp(1j*phi)*(b2*t2*exp(1j*phi)+a3*1j*k2), a3 == b3*exp(2j*psi));

Warning: Unable to find explicit solution. For options, see help.

S.b1

 
ans =
 
Empty sym: 0-by-1
 

However, I'd like to try to plot each variable (say, b1) with respect to phi and psi changing. This means that I can't define it before hand. I've hit a bit of a roadblock as to how to do this.

Answer 1

[Walter Roberson]

t = 1;
t2 = 1;
k = 1;
k2 = 1;
a1 = 1;
syms a2 a3 b1 b2 b3 phi psi
eqns = [b1 == a1*t + a2*1j*k,b2 == a1*1j*k + a2*t, b3==a3*t2 + b2*1j*k2*exp(1j*phi), ...
    a2 == exp(1j*phi)*(b2*t2*exp(1j*phi)+a3*1j*k2), a3 == b3*exp(2j*psi)]

eqns = 

symvar(eqns)

ans = 

size(eqns)

ans = 1×2

1 5

S = solve(eqns)

Warning: Unable to find explicit solution. For options, see help.

S = struct with fields:
     b1: [0×1 sym]
     b2: [0×1 sym]
     b3: [0×1 sym]
    phi: [0×1 sym]
    psi: [0×1 sym]

S.b1

 
ans =
 
Empty sym: 0-by-1
 

You have 5 equations in 6 unknowns. solve() is going to guess which variables to solve for

S2 = solve(eqns, [a2 a3 b1 b2 b3])

S2 = struct with fields:
    a2: -(exp(phi*2i)*1i - exp(phi*2i)*exp(psi*2i)*2i)/(exp(phi*2i) + exp(psi*2i) - 2*exp(phi*2i)*exp(psi*2i) - 1)
    a3: (exp(phi*1i)*exp(psi*2i))/(exp(phi*2i) + exp(psi*2i) - 2*exp(phi*2i)*exp(psi*2i) - 1)
    b1: (2*exp(phi*2i) + exp(psi*2i) - 4*exp(phi*2i)*exp(psi*2i) - 1)/(exp(phi*2i) + exp(psi*2i) - 2*exp(phi*2i)*exp(psi*2i) - 1)
    b2: (exp(psi*2i)*1i - 1i)/(exp(phi*2i) + exp(psi*2i) - 2*exp(phi*2i)*exp(psi*2i) - 1)
    b3: exp(phi*1i)/(exp(phi*2i) + exp(psi*2i) - 2*exp(phi*2i)*exp(psi*2i) - 1)

[Dhruv Srinivasan]

Thank you so much for your help!