When to stop looking for closed-form solutions?

Question

I have the following equation which is simplified from another relatively long equation. $$ xe^{-x}= c (\frac{x}{c})^{-x} $$ Here Lambert W function doesn't help since you still get $x$ inside the Lambert W.

I know that I can use numerical methods to find the interception(s) of these two lines. But in general, I wonder at what point we can realize that there's no hope of finding the closed-form solution and we should stop looking for it.

Answer 1

Assume that both $x$ and $c$ are positive. So, as @user619894 commented $$x^{1+x}e^{-x}=c^{1+x} \quad \implies \quad c=x \,e^{-\frac{x}{x+1}}\tag 1$$

$$\frac{x}{x+1}=y\quad \implies \quad c=\frac{e^{-y} \,y}{1-y}\quad \implies \quad \color{red}{e^{-y}=c \,\frac {1-y}y}$$ which has an explicit solution in terms of the generalized Lambert function (have a look at equation $(4)$).

This being said, if it is to have obtained a closed form expression, from a practical point of view, it is not very useful. However, if we consider the case of large values of $x$ and $c$, we can write

$$x \,e^{-\frac{x}{x+1}}=\frac 1e \Big[x+1+\sum_{n=1}^\infty \frac {a_n}{(n+1)! \,x^n}\Big]$$ The coefficient form sequence $A111884$ in $OEIS$.

Truncated to some order and using series reversion (with $t=e\,c$) $$x=t-1+\frac{1}{2 t}+\frac{1}{3 t^2}-\frac{1}{8 t^3}-\frac{7}{15 t^4}-\frac{31}{144 t^5}+\frac{19}{35 t^6}+\frac{4927}{5760 t^7}-\frac{563}{2835 t^8}+O\left(\frac{1}{t^9}\right)$$

Using the above terms for $c=10$ gives the estimate $x=\color{red}{26.20165628306}409$ to be compared to the solution obtained by Newton method is $x=\color{red}{26.20165628306386}$

Even for $c=1$, the above truncated series gives $x=1.93317$ to bee compared to the solution $x=1.93295$.

Reinjecting this expression in $(1)$, the difference between lhs and rhs is $\frac{81479}{44800 e^{10} c^9} \sim \frac{8.25\times 10^{-5} } {c^9}$

Answer 2

You have to decide why you prefer a closed-form solution. Sometimes it's sufficient to give your solution a name and say that it is in closed form then. But if you want to calculate with your closed-form solutions or wants to compare them, you need the methods for calculating in your set of closed-form functions/numbers.
Finding a solution in closed form means you get hints about series presentations, calculation rules, properties etc.

You have to decide which kinds of constants, operations/functions and variables you want to accept to be allowed to call it in closed form. Simple classes of expressions e.g. are radical expressions (explicit algebraic expressions) or elementary functions.
Further classes of functions could be e.g. the algebraic functions, Lambert W, generalizations of Lambert W (e.g. as the previous answer suggests), special standard functions, hypergeometric functions, generalized hypergeometric functions.

You can decide to stop if you have checked all known classes of functions.