Solving for a variable in matlab - matlab

Question

I have a system of two equations and I need Matlab to solve for a certain variable. The problem is the variable I need is inside an expression, and trig functions. I wrote the following code:
function [ V1, V2 ] = find_voltages( w1, l1, d, w2, G1, G2, m, v, e, h, a, x)
k1 = sqrt((2*V1*e)/(G1^2*m*v^2));
k2 = sqrt((2*V2*e)/(G2^2*m*v^2));
A = h + l1*a;
b = -A*k1*sin(k1*w1) + a*cos(k1*w1);
B = A*cos(k1*w1) + (a/k1)*sin(k1*w1);
C = B + a*b;
c = C*k2*sinh(k2*w2) + b*cosh(k2*w2);
D = C*cosh(k2*w2) + (b/k2)*sinh(k2*w2);
bd = A*k1*sinh(k1*w1) + a*cosh(k1*w1);
Bd = A*cosh(k1*w1) + (a/k1)*sinh(k1*w1);
Cd = Bd + a*bd;
cd = -Cd*k2*sin(k2*w2) + bd*cos(k2*w2);
Dd = Cd*cos(k2*w2) + (bd/k2)*sin(k2*w2);
fsolve([c*(x-(l1+w1+d+w2)) + D == 0, cd*(x-(l1+w1+d+w2)) + Dd == 0], [V1,V2])
end
and got an error because V1 and V2 are not defined. They are part of an expression, and need to be solved for. Is there a way to do this? Also, is it a problem that the functions I put as arguments to solve are conglomerates of the smaller equations above them?
Valid values:
Drift space 1 (l1): 0.11
Quad 1 length (w1): 0.11
Quad 2 length (w2): 0.048
Separation (d): 0.014
Radius of Separation 1 (G1): 0.016
Radius of Separation 2 (G2): 0.01
Voltage 1 (V1): -588.5
Voltage 2 (V2): 418
Kinetic Energy in eV: 15000
Mass (m) 9.109E-31
Kinetic Energy in Joules (K): 2.4E-15
Velocity (v): 72591415.94
Charge on an Electron (e): 1.602E-19
k1^2=(2*V1*e)/(G1^2*m*v^2): 153.4467773
k2^2=(2*V2*e)/(G2^2*m*v^2): 279.015

Answer 1

First start by re-writing your function as an expression which returns the extent to which your function(s) fail to hold for some valid guess for [V1,V2]. E.g.,
function gap = voltage_eqn(V, w1, l1, d, w2, G1, G2, m, v, e, h, a, x)
V1 = V(1) ;
V2 = V(2) ;
k1 = sqrt((2*V1*e)/(G1^2*m*v^2));
k2 = sqrt((2*V2*e)/(G2^2*m*v^2));
A = h + l1*a;
b = -A*k1*sin(k1*w1) + a*cos(k1*w1);
B = A*cos(k1*w1) + (a/k1)*sin(k1*w1);
C = B + a*b;
c = C*k2*sinh(k2*w2) + b*cosh(k2*w2);
D = C*cosh(k2*w2) + (b/k2)*sinh(k2*w2);
bd = A*k1*sinh(k1*w1) + a*cosh(k1*w1);
Bd = A*cosh(k1*w1) + (a/k1)*sinh(k1*w1);
Cd = Bd + a*bd;
cd = -Cd*k2*sin(k2*w2) + bd*cos(k2*w2);
Dd = Cd*cos(k2*w2) + (bd/k2)*sin(k2*w2);
gap(2) = c*(x-(l1+w1+d+w2)) + D ;
gap(1) = cd*(x-(l1+w1+d+w2)) + Dd ;
end
Then call fsolve from some initial V0:
Vf = fsolve(#(V) voltage_eqn(V, w1, l1, d, w2, G1, G2, q, m, v, e, h, a, x), V0) ;