How to solve or find approximation for solution

Question

How this equation can be solved: $xe^{x/a}-b=0$ ($a$,$b$,$x$ assumed as real numbers)? Namely, I need to find solution in analytical form. Maybe there are some methods to approximate this equation and than express x from this approximation? I need any ideas. Thank you!

Answer 1

$x\cdot e^{\frac xa} = b$

$\implies \frac xa\cdot e^{\frac xa}= \frac ba $

The solution is given by the Lambert W function,

$ie \qquad\frac{x}{a} = W\bigg(\frac ba\bigg)$

$x= a\cdot W\bigg(\frac{b}{a}\bigg)$

You can approximate the argument with ;

$W(x) = \sum_{n=1}^\infty \dfrac{(-n)^{n-1}}{n!}x^n=x-x^2+\frac32x^3-\frac83x^4+\cdots$

You could also use Newtons method to find the roots with iteration ;

Define $h(x) = x\cdot e^{\frac xa}-b$

$\implies h'(x)=e^{\frac xa}+\frac xa\cdot e^{\frac xa} $

$x= x_0-\frac{h(x_0)}{h'(x_0)}$

$x = x_0- \frac{x\cdot e^{\frac xa}-b}{e^{\frac xa}+\frac xa\cdot e^{\frac xa}}$

With $a= 4 ,b =2,x_0 =1$, i found that the answer quickly converges to $1.406935$ after $4$ interations