# Solving complex equation.

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Abdullah Alsayegh on 22 Feb 2022
Commented: Walter Roberson on 23 Feb 2022
I want to create an equation in matlab and for each one variable that is unknown it can solve it directly.
For example I sign every variable to equal a number and solve for Cj.
Below is my coding to solve for Cj.
However I want to create an equation that let say if I have Cj known and I want to find H or y it can solve it without me having to creat another equation to solve for it
mdot = 115; % Source Strength g/s
U = 6.82; % Wind speed m/s
h = 62.6; % Stack Height (m)
Sh = 25.5; % Plume Rise Height (m)
H = h + Sh; % Effective Stack Height (m)
x = 2; % X location (m)
y = 0; % Y location (m)
z = 1.49; % Z location (m)
% Dispersion Coefficients
a = 68;
b = 0.894;
c = 44.5;
d = 0.516;
f = -13;
% Dispersion Coefficients
Sigma_Y = a * (x^b);
Sigma_Z = c * (x^d) + f;
% Exp
e1 = exp((-1/2)*((y/Sigma_Y)^2 + ((z-H)/Sigma_Z)^2));
e2 = exp((-1/2)*((y/Sigma_Y)^2 + ((z+H)/Sigma_Z)^2));
% Equation for Gaussian concentration for absorbing ground
Cj1 = (mdot * e1) / (2*pi*U*Sigma_Y*Sigma_Z);
Cj_Absorbing = Cj1 * 10^6;
% Equation for Gaussian concentration for reflective ground
Cj1 = (mdot * (e1 + e2)) / (2*pi*U*Sigma_Y*Sigma_Z);
Cj_Reflecting = Cj1 * 10^6;
Torsten on 22 Feb 2022
syms x y
expr = 3*x^2 + sin(y) == 0;
solx = solve(expr,x);
solx = double(subs(solx,y,0.3))
soly = solve(expr,y);
soly = double(subs(soly,x,0.2))

David Hill on 22 Feb 2022
Edited: David Hill on 22 Feb 2022
function Output = solveE(Input,guess)
%Input is an array [U,mdot,H,z,y,sigmaY,sigmaZ,Cj] of all values except nan
%for the variable you want to solve for.
syms U mdot H y z sigmaY sigmaZ Cj
eqn=mdot/(2*pi*U*sigmaY*sigmaZ)*exp(-0.5*((y/sigmaY)^2+((z-H)/sigmaZ)^2))-Cj==0;
switch find(isnan(Input))
case 1
Output=vpasolve(subs(eqn,[mdot,H,z,y,sigmaY,sigmaZ,Cj],Input(2:8)),U,guess);
case 2
Output=vpasolve(subs(eqn,[U,H,z,y,sigmaY,sigmaZ,Cj],Input([1,3:8])),mdot,guess);
case 3
Output=vpasolve(subs(eqn,[U,mdot,z,y,sigmaY,sigmaZ,Cj],Input([1:2,4:8])),H,guess);
case 4
Output=vpasolve(subs(eqn,[U,mdot,H,y,sigmaY,sigmaZ,Cj],Input([1:3,5:8])),z,guess);
case 5
Output=vpasolve(subs(eqn,[U,mdot,H,z,sigmaY,sigmaZ,Cj],Input([1:4,6:8])),y,guess);
case 6
Output=vpasolve(subs(eqn,[U,mdot,H,z,y,sigmaZ,Cj],Input([1:5,7:8])),sigmaY,guess);
case 7
Output=vpasolve(subs(eqn,[U,mdot,H,z,y,sigmaY,Cj],Input([1:6,8])),sigmaZ,guess);
case 8
Output=vpasolve(subs(eqn,[U,mdot,H,z,y,sigmaY,sigmaZ],Input(1:7)),Cj,guess);
end
Walter Roberson on 23 Feb 2022
? your equation has no x, and your code has no x, so I do not see where you would put x=1:1:10 ?
If you are doing symbolic work and you put a vector into vpasolve() or solve() then those functions need to solve the set of equations simultaneously . For example if you had 3*x == z and you wanted to know z and you put in x = [1, 2] then it would be trying to solve [3 == z, 6 == z] simultaneously , which would fail. vpasolve() and solve() are for simultaneous equations .