You write,
This is incorrect. K = 1.38064852*10^(-23) +/- 79 * 10^(-30).
You have asked for an exact solution to a calculation using incorrect values -- notice that even if you only used 5 significant digits then you should have written K=1.3806*10^-23 not K=1.3807*10^-23
I could go through and pick on your other floating point constants as well, and I could point out that MATLAB's internal representation of what you wrote would in fact be K = 1.380699999999999892724734311718274525674787081853034352153149493143413639728578345966525375843048095703125e-23 .
What you need to understand from this is that it is almost always an error to ask for exact solutions to any equation in which you have used even one floating point number.
Also, you have
and you use that value, and then later you have
syms Vs Ga y k1 e x Theta nstar Nb Pb Ld ub
eqns=-e/(1+(Ga*exp(-y/k1))*(exp(x/k1)*exp(-(e*Vs(1)/k1))*Theta*nstar==sqrt(2)*(Nb+Pb)*e*Ld*sqrt((cosh(ub+(e*Vs))/k1))/(cosh(ub-(e*Vs)))/(k1)*tanh(ub)-1))
Vs= solve(eqns, Vs)
in which you confusingly use the same variable name Vs .
Notice that your equation includes exp(-(e*Vs(1)/k1)) . MATLAB would resolve that to Vs because that that point Vs would be a scalar symbol, but it is confusing to the reader. Are you expecting Vs to be solved to become a vector and at that particular point you only want the first element of the vector? Did some characters go missing between the Vs and the (1) ?
If you make the assumption that each constant represents an exact decimal, such as assuming that 2.42*1.6*10^-19; represents exactly (242/100)*(16/10)*1/(10^19), and work in the symbolic toolbox, then if you take the left hand side of eqns minus the right hand side of eqns, turning in into an expression in a single variable, you will find that at no point does the difference become zero. The limit as the single variable approaches -infinity is
1/(6250000000000000000*(1-2*exp(-6400/207)))
which is about 1.60000000000011958803665930542*10^(-19) . The limit as the single variable approaches +infinity is
exp(-6400/207)/(3125000000000000000*(1-2*exp(-6400/207)))
which is about 1.19588036659305418113895190508*10^(-32) . Both of these are positive and the expression is monotonically decreasing, so the expression does not pass through zero.
Therefore under the assumption that the constants given are completely correct, there are no solutions to the equation.
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