How to solve polynomial-exponential equation

Question

I'm trying to solve equations like the following one:

$$5 + 3x - 4x^3 = e^{x^2}$$

I've tried using the Lambert W function, but I didn't get any success. I must admit I'm relatively new to Lambert W function.

I've managed to bring the equation to:

$$e^{-x^2}(5 + 3x - 4x^3) = 1$$

But I can't bring the exponent and the second term on the LHS to the same value, in order to apply Lambert W function.

Also I tried to search on the Internet, but I never found a equation from a simular type, all of them were from the type:

$$5^x = 7x$$

This type of equation is fairly easy to solve using Lambert W function, but it doesn't help me solving an equation from the first type.

Also I couldn't come up with another idea how to get solution, except for Lambert W function. Can you please help me?

Answer 1

There is no really fundamental difference between the function $$ f(x) = x e^x$$ and your $$g(x) = (5+3x-4x^3)e^{-x^2}$$.

The only point is that the first function is famous that the inverse $f^{-1}$ has an name ("Lambert W function"). But its just a name. Mathematically, the situation is essentially the same:

• There is no nice closed expression (say, using only $+,-,\times,/,\sqrt{},\sin,\cos,\exp$, or whatever you would consider a simple expression).
• But its easy enough to find any value numerically.
• Also you can derive Taylor-expansions, various limits, and plenty of properties of (for example) the derivatives of $g^{-1}$.

This situation is actually very common in math. There are plenty of functions (actually most) that exist perfectly well, but cant be written down in a nice closed form. Some are famous enough to get a name ("Gamma function", "Logarithmic integral function", "Pochhammer-symbol", "hypergeometric functions", "elliptic integral", and many many more), these are usually called "special functions". But just giving something a name doesnt really do anything mathematically. And just because you cant write down a nice equation for a function doesnt mean it is any less real.

Answer 2

$$5+3x-4x^3=e^{x^2}$$

We see, this equation is a polynomial equation of more than one algebraically independent monomials ($x,e^{x^2}$) and with no univariate factor. We therefore don't know how to rearrange the equation for $x$ by applying only finite numbers of elementary functions (operations) we can read from the equation.

Because your equation can be rearranged to a polynomial exponential equation over the algebraic numbers, it cannot have solutions that are elementary numbers.

Because your exponential polynomial equation has a polynomial term of more than one non-constant monomials of different degree, Lambert W cannot applied here.

But the equation can possibly be solved by Generalized Lambert W - see the references below. $\ $

[Mezö 2017] Mezö, I.: On the structure of the solution set of a generalized Euler-Lambert equation. J. Math. Anal. Appl. 455 (2017) (1) 538-553

[Mezö/Baricz 2017] Mezö, I.; Baricz, A.: On the generalization of the Lambert W function with applications in theoretical physics. Transact. Amer. Math. Soc. 369 (2017) (11) 7917–7934

[Castle 2018] Castle, P.: Taylor series for generalized Lambert W functions. 2018