How do I plot the relationship between two variables in an inseparable function?

I am trying to plot the relationship between the variables S and t in the following equation:

S = 8-0.7*t+2.5*log(8/S)

However, I do not know how to create a plot of S vs t given that the function is inseparable. What is the syntax in Matlab for plotting a function of more than one variable?

Answer 1

[Renato Medeiros]

I guess you are trying to solve the implicit equation S = 8-0.7*t+2.5*log(8/S) and plot the result. If this is the case, you could solve for S after you determine a range for t. For this, you can use the function fsolve, passing the specified t as a parameter at each new solution, like:

function answer
%solve implicit function and plot
npoints = 100;

t = linspace(0.1,10,npoints);
S = zeros(1,npoints);

guess = 10;

for i = 1:npoints
    S(i) = fzero(@equation,guess,[],t(i));
    guess = S(i);
end

plot(t,S)
xlabel('t')
ylabel('S')

function dy = equation(S,t)
dy = 8-0.7*t+2.5*log(8/S) - S;

[Dimitri Kaviani]

Can you explain the syntax of the fzero command? I understand that you are required to input the function and an initial guess but what are the other inputs you have listed above?

[Walter Roberson]

Renato is showing an undocumented syntax of fzero that could go away.

The call should be

S(i) = fzero(@(s) equation(s,t(i)), guess);

Answer 2

[Massimo Zanetti]

(Erroneously canceled my previous answer) Not all s,T verify the equation. To find them use fsolve https://it.mathworks.com/help/optim/ug/fsolve.html

To just plot, you need to see the equation as a surface of t,s as follows:

t=-2:.1:2;
s=0:.1:5;
[T,S]=meshgrid(t,s);
F = 8-0.7*T+2.5*log(8./S)-S;
surf(T,S,F);
xlabel('t');
label('S');

You can also look at the points that verify the equation by

contour(F,[0,0]);

Answer 3

[Walter Roberson]

The equations are separable:

t = 80/7+(25/7)*ln(8/S)-(10/7)*S

or

S = (5/2)*LambertW((16/5)*exp(16/5-(7/25)*t))

There are additional complex solutions for S, all of the other branches of LambertW; you did not specify the solution domain

You can also plot directly:

ezplot(@(t,S) -S + 8 - 0.7 .* t + 2.5 .* log(8./S), [-10 10])