I need help with solving this difficult fluid dynamic expression. I have tried using rules of logs, symbolab algebra calculator and Wolfram Alpha calculator, and I have got no solution.
How would you solve the following expression for $x$?
$$e^{\frac{x^2}{4vt}} = 1+\frac{x^2}{2vt}$$
When solving this numerically, the solution is: $x=2.2418\sqrt{vt}$
I want to know how you could solve the first expression to get the solution. So could someone please provide a step-by-step solution please?
Best Answer
$$e^{\frac{x^2}{4vt}} = 1+\frac{x^2}{2vt}$$ Let $y=\frac{x^2}{4vt}$ $$e^y=1+2y$$ $$e^{-y}=\frac{1}{1+2y}$$ $$(1+2y)e^{-y}=1$$ $$(\frac12+y)e^{-y}=\frac12$$ $$(-\frac12-y)e^{-y}=-\frac12$$ $$ (-\frac12-y)e^{-y}e^{-\frac12}=-\frac12 e^{-\frac12}$$ $$ (-\frac12-y)e^{-\frac12-y}=-\frac12 e^{-\frac12}=-\frac{1}{2\sqrt{e}}$$ $X=(-\frac12-y)$ $$Xe^X=-\frac{1}{2\sqrt{e}}$$ From the definition of the Lambert W function : http://mathworld.wolfram.com/LambertW-Function.html $$X=W\left(-\frac{1}{2\sqrt{e}}\right)$$ $$y=-\frac12-X=-\frac12-W\left(-\frac{1}{2\sqrt{e}}\right)$$ $$x=\sqrt{4vty}=\sqrt{-2-4W\left(-\frac{1}{2\sqrt{e}}\right)}\sqrt{vt}$$ The Lambert W(z) function is a multi valuated function in $-\frac{1}{e} <z<0$ , real $z$.
This is presently the case where $z=-\frac{1}{2\sqrt{e}}$ since $-\frac{1}{e} <-\frac{1}{2\sqrt{e}}<0$
First root :
$W_0\left(-\frac{1}{2\sqrt{e}}\right)=-\frac12 \quad;\quad {-2-4W_0\left(-\frac{1}{2\sqrt{e}}\right)}=-2-4(-1/2)=0 \quad;\quad x=0$
Second root :
$W_{-1}\left(-\frac{1}{2\sqrt{e}}\right)\simeq -1.756431... $
https://www.wolframalpha.com/input/?i=lambertw(-1,-1%2F(2+sqrt(e)))
One can use series expansion to compute it approximately. See page 13 in https://fr.scribd.com/doc/34977341/Sophomore-s-Dream-Function . The convergence is very slow. See the numerical calculus in the below addition. In practice, it is certainly faster to solve directly $e^y=1+2y$ with Newton-Raphson or similar iterative method.
Finally, an approximate value is :
$$x\simeq\sqrt{-2-4(-1.756431)}\sqrt{vt}\simeq 2.2418128 \sqrt{vt}$$
With more digits : https://www.wolframalpha.com/input/?i=sqrt(-2-4+lambertw(-1,-1%2F(2+sqrt(e))))
IN ADDITION :
Example of recursive numerical calculus of $W_{-1}(x)$ in the range $-\frac{1}{e}<x<0$