How do you use solve() to solve a complicated symbolic equation with one variable in terms of unknown constants?

I have a a complicated symbolic equation that I would like to solve for x in terms of the letter-name constants. I received the following warning:

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

In solve (line 317)

solx =

Empty sym: 0-by-1

This is what I input:

syms x a b c d e f g
eqn = a*x - ((b^2)/2)*(a/b)*((1 - 4*exp(-2*(c-x)/b))^(-1)) - b*(a/b)*((1 - 4*exp(-2*(c-x)/b))^(-1))*(x+(((1/b)+d)^(-1))*log(1-(((b/(g*e))+(g/f))*((b/a)^(-0.5)))*((1-4*exp(-2*(c-x)/b))^(-0.5)))) - ((d*e)/2)*(((1/b)+((2*d)/b)+d^2)^(-1))*(1-2*exp((((1/b)+d)^(-1))*log(1-((b/(g*e))+(g/f))*((b/a)^(-0.5))*((1-4*exp(-2*(c-x)/b))^(-0.5)))*((1/b)+d)) + exp(2*(((1/b)+d)^(-1))*log(1 - ((b/g*e)+(g/f))*((b/a)^(-0.5))*((1-4*exp(-2*(c-x)/b))^(-0.5)))*((1/b)+d)) - (((1/b)+d)^(-1))*log(1 - ((b/(g*e))+(g/f))*((b/a)^(-0.5))*((1 - 4*exp(-2*(c-x)/b))^(-0.5)))*(1/b) - (2/b)*(((1/b)+d)^(-1)) + (2/b)*(((1/b)+d)^(-1))*exp((((1/b)+d)^(-1))*log(1 - ((b/(g*e))+(g/f))*((b/a)^(-0.5))*((1-4*exp(-2*(c-x)/b))^(-0.5)))) + (1/(2*b))*(((1/b)+d)^(-1)) - (1/(2*b))*(((1/b)+d)^(-1))*exp(2*(((1/b)+d)^(-1))*log(1-((b/(g*e))+(g/f))*((b/a)^(-0.5))*((1-4*exp(-2*(c-x)/b))^(-0.5)))*((1/b)+d)) == 0;
solx = solve(eqn, x)

Is there a way to proceed to a solution for x in terms of the letter-name constants? I have already used the simplify(eqn) command and several assume(x>0), assume(e>0), etc. parameters based on what I can assume from the physical constants, but to no avail.

[Walter Roberson]

There seems to be a ') miscount in your eqn.

[Kelly Catlin]

Easy to make such an error with this many parentheses. I think a corrected version should be:

eqn = (a*x - ((b^2)/2)*(a/b)*((1 - 4*exp(-2*(c-x)/b))^(-1)) - b*(a/b)*((1 - 4*exp(-2*(c-x)/b))^(-1))*(x+(((1/b)+d)^(-1))*log(1-(((b/(g*e))+(g/f))*((b/a)^(-0.5)))*((1-4*exp(-2*(c-x)/b))^(-0.5)))) - ((d*e)/2)*(((1/b)+((2*d)/b)+d^2)^(-1))*(1-2*exp((((1/b)+d)^(-1))*log(1-((b/(g*e))+(g/f))*((b/a)^(-0.5))*((1-4*exp(-2*(c-x)/b))^(-0.5)))*((1/b)+d)) + exp(2*(((1/b)+d)^(-1))*log(1 - ((b/(g*e))+(g/f))*((b/a)^(-0.5))*((1-4*exp(-2*(c-x)/b))^(-0.5)))*((1/b)+d)) - (((1/b)+d)^(-1))*log(1 - ((b/(g*e))+(g/f))*((b/a)^(-0.5))*((1 - 4*exp(-2*(c-x)/b))^(-0.5)))*(1/b) - (2/b)*(((1/b)+d)^(-1)) + (2/b)*(((1/b)+d)^(-1))*exp((((1/b)+d)^(-1))*log(1 - ((b/(g*e))+(g/f))*((b/a)^(-0.5))*((1-4*exp(-2*(c-x)/b))^(-0.5)))) + (1/(2*b))*(((1/b)+d)^(-1)) - (1/(2*b))*(((1/b)+d)^(-1))*exp(2*(((1/b)+d)^(-1))*log(1-((b/(g*e))+(g/f))*((b/a)^(-0.5))*((1-4*exp(-2*(c-x)/b))^(-0.5)))*((1/b)+d)) == 0));

[Walter Roberson]

Are you sure? The system looks to me as if it should be a standard

expression == 0

but you have something that ends in

... something == 0))

implying that the true or false result of the "==" is to be taken as 1 or 0 and used in computation of an expression. If that is what you want, then that would need to be rewritten for symbolic work.

[Kelly Catlin]

I agree. I do not intend to have a TF system, yet when I enter:

eqn = a*x - ((b^2)/2)*(a/b)*((1 - 4*exp(-2*(c-x)/b))^(-1)) - b*(a/b)*((1 - 4*exp(-2*(c-x)/b))^(-1))*(x+(((1/b)+d)^(-1))*log(1-(((b/(g*e))+(g/f))*((b/a)^(-0.5)))*((1-4*exp(-2*(c-x)/b))^(-0.5)))) - ((d*e)/2)*(((1/b)+((2*d)/b)+d^2)^(-1))*(1-2*exp((((1/b)+d)^(-1))*log(1-((b/(g*e))+(g/f))*((b/a)^(-0.5))*((1-4*exp(-2*(c-x)/b))^(-0.5)))*((1/b)+d)) + exp(2*(((1/b)+d)^(-1))*log(1 - ((b/g*e)+(g/f))*((b/a)^(-0.5))*((1-4*exp(-2*(c-x)/b))^(-0.5)))*((1/b)+d)) - (((1/b)+d)^(-1))*log(1 - ((b/(g*e))+(g/f))*((b/a)^(-0.5))*((1 - 4*exp(-2*(c-x)/b))^(-0.5)))*(1/b) - (2/b)*(((1/b)+d)^(-1)) + (2/b)*(((1/b)+d)^(-1))*exp((((1/b)+d)^(-1))*log(1 - ((b/(g*e))+(g/f))*((b/a)^(-0.5))*((1-4*exp(-2*(c-x)/b))^(-0.5)))) + (1/(2*b))*(((1/b)+d)^(-1)) - (1/(2*b))*(((1/b)+d)^(-1))*exp(2*(((1/b)+d)^(-1))*log(1-((b/(g*e))+(g/f))*((b/a)^(-0.5))*((1-4*exp(-2*(c-x)/b))^(-0.5)))*((1/b)+d)) == 0;

I received the error: "Unbalanced or unexpected parenthesis or bracket" with an arrow pointed to the semicolon beyond the zero. The suggestion "did you mean..." then provides the suggestion that is identical to the above, save with .... == 0);. Yet this is not what I intend. I have checked the parentheses in my above eqn, and there should be no imbalance.

[Walter Roberson]

I gave up marking out nesting for you when I hit nesting level 12. You should break up your equation into smaller parts. If you go past roughly nesting level 5 then it has probably become too difficult for humans to easily understand.

[Kelly Catlin]

Could you explain precisely what you mean by that? Do you mean simplify the expression by substituting more terms, e.g...

eqn = a*x - ((b^2)/2)*(a/b)*j - b*(a/b)*j*(x+(((1/b)+d)^(-1))*h) - ((d*e)/2)*(((1/b)+((2*d)/b)+d^2)^(-1))*(1-2*exp((((1/b)+d)^(-1))*h*((1/b)+d)) + exp(2*(((1/b)+d)^(-1))*h*((1/b)+d)) - (((1/b)+d)^(-1))*h*(1/b) - (2/b)*(((1/b)+d)^(-1)) + (2/b)*(((1/b)+d)^(-1))*exp((((1/b)+d)^(-1))*h*((1/b)+d)) + (1/(2*b))*(((1/b)+d)^(-1)) - (1/(2*b))*(((1/b)+d)^(-1))*exp(2*(((1/b)+d)^(-1))*h *((1/b)+d)) == 0;

[Kelly Catlin]

Scratch that. I've modified the defining of the constants so that there are far fewer terms:

eqn3 = E + a*x - b + 2*b*exp(-x*c) - b*exp(-2*x*c) - (x/d)*b - (b/d)*2*(1/c)*exp(-x*c) + (b/d)*2*(1/c) - (1/(2*d))*b*(1/c) + (1/(2*d))*b*(1/c)*exp(-2*x*c) == 0;

There is no longer any errors for unbalanced parentheses, and solx = solve(eqn3,x) executed. Yet I am now receiving a warning:

Warning: Unable to find explicit solution. For options, see help. > In solve (line 317)

solx =

Empty sym: 0-by-1

Does this mean that the equation cannot be solved explicitly?

[Walter Roberson]

Basically, I was talking about using more temporary variables to define parts of the equation.

For example if you are constructing the equation term by term, then put each term into a different variable and then finally add them to get the final equation. This would make it much easier to read and to figure out why the ) were not matching.

[Walter Roberson]

There is no closed-form solution for eqn3 .

The sum of exponentials does not have a general closed form solution. Sometimes equations with exponentials can be solved with LambertW, but not generally.

[Kelly Catlin]

You are correct, LambertW does not appear to work with this equation. I attempted Newton's Method, instead, but of course having a largely symbolic equation makes numerical approximation difficult. Is there even a script for approximating with Newton's Method when constants are not explicitly defined?

[Walter Roberson]

"Is there even a script for approximating with Newton's Method when constants are not explicitly defined?"

At that point what you are heading towards is doing a taylor series and solve()'ing what results.

In Maple the sequence would be

eqn3 := E+a*x-b+2*b*exp(-x*c)-b*exp(-2*x*c)-x*b/d-2*b*exp(-x*c)/(d*c)+2*b/(d*c)-b/((2*d)*c)+b*exp(-2*x*c)/((2*d)*c);
convert(taylor(eqn3, x = 0, 5), polynom);
solve(%);
simplify([allvalues](%),size) assuming nonnegative;

which would get you a wall of about 75 lines of unreadable closed-form solution. And since this is only degree 5 approximation, the approximation to the exponential is not going to be all that good, especially further away from x = 0. You cannot go for any higher degree because you would get a quintic or higher degree, for which there are no closed form solutions. Even degree 4 should only be treated as a vague approximation, perhaps a starting point for numeric solutions.