Unable to find explicit solution, but I know a solution exists

Hi,

I am solving a complex equation using the following code:

syms x h k l m n b c CTL eps_y g
eq = 2*((h.*x.^2)./eps_y).^(1/4) + m.*(((g.^(4*n)).*x.^(2*k*(1-n)+2*n*(l+2)))./((h.*eps_y^3).^2)).^(1/(4*k))...
    == 2*x./(eps_y*CTL) + m.*(x.*g.*CTL^(k/(n-1))).^(n/k)
eq = rewrite(eq,'log');
solx = solve(eq, x, 'IgnoreAnalyticConstraints',1)
pretty(solx)

I know that an analytical solution exists, because when I plot the two functions I am equating w.r.t the variable x, they are intersecting. The paper I am following says the solution to this is the following:

I used Π = x and =CTL to make things simpler. My full code has all the values except for x, but I want a solution that is in terms of all the symbols. I also did try subbing in numbers that I know result in an intersection and it still cannot find a solution. Any suggestions what I can try?

[Walter Roberson]

The solution is x = 0.

Being able to see an intersection in a plot does not mean that there is an analytical solution, not even if the intersection turns out to be exactly at an integer.

[Ayman Al-Sukhon]

Hi Walter,

Thanks for your fast answer.

Perhaps my math skills are not as sharp as I thought. How can it be possible there is an intersection and yet no analytical solution? There exists a value of Π where the equations on each side intersect, doesn't there?

Edit: Interestingly enough, it seems that upon further inspection of the paper, what you are saying is actually baked into the physics of the problem from the paper I am following. I just wonder how this is possible mathematically...

[Walter Roberson]

When people talk about a problem being analytic, they usually mean that there is a closed form formula in "algebraic numbers", possibly extended with trig and LambertW and some other formulas. Sometimes they include some of the hypergeometric functions, but that varies. Generally speaking, some closed formula that provides explicit ways to find the solution to any desired precision.

However there are a lot of functions that are not especially difficult to draw but there is no known explicit formula to solve. For example sin(x) == exp(x) has an infinite number of solutions but there is no known formula to find them.

[John D'Errico]

Lambertw really is the perfect example.

syms x,
solve(x + exp(x) == 2,x)
ans =
2 - lambertw(0, exp(2))

So a solution exists, but only because a special function was created to yield one. There is no other algebraic solution, thus only in terms of radicals and elementary operations.

This means that even though you can see a solution, there may be no way to compute it analytically.

Another great example would be in the form of some 5th (or higher) degree polynomial.

syms x
P = randi(5,[1 6])
P =
     1     1     2     5     1     5

P = dot(P,x.^(5:-1:0))
P =
x^5 + x^4 + 2*x^3 + 5*x^2 + x + 5
 
vpa(solve(P))
ans =
 - 0.026642852245247853801004512805907 - 1.164802485283373680594682049458i
 - 0.026642852245247853801004512805907 + 1.164802485283373680594682049458i
   0.41584127815813601809947113687271 - 1.3777578385788797695485053192232i
   0.41584127815813601809947113687271 + 1.3777578385788797695485053192232i
                                        -1.7783968518257763285969332481336

P(x) has roots, exactly one of which is real. However we will not be able to write them in an algebraic form. We can clearly see them in a plot.

fplot(P,[-3,2])
yline(0);

But does that mean you can find an analytical solution for the roots? Sadly, Abel-Ruffini suggests we are most likely out of luck. You can see that real root right there though, clear as day, yet frustratingly unobtainable.

Finally, it is also sometimes the case where a solution DOES exist, yet it requires some intelligent mathematics to extract. I recall Cleve Moler writing a blog recently, where he manages to extract a solution for a problem, despite the failure of a brute force call to solve. Solvers are not perfect and omniscient. Sometimes they need some creative guidance.

[Walter Roberson]

For that matter, logarithm is not strictly considered analytic, in the sense of not being just "algebraic numbers". e is not an algebraic number.

On the other hand, https://machinelearningmastery.com/analytical-vs-numerical-solutions-in-machine-learning/ describes

• An analytical solution involves framing the problem in a well-understood form and calculating the exact solution.
• A numerical solution means making guesses at the solution and testing whether the problem is solved well enough to stop.

Something like logorithm involves an exact solution with well-understood calculation methods. But my example of sin(x)==exp(x) requires just guessing and narrowing in on correct values even though the graph is simple enough.