How to solve $xe^x+e^x= e^4$?

Question

How to solve $xe^x+e^x= e^4$?

I have come across this problem. But I can't solve this problem. I would be very grateful if someone helped me.

Answer 1

This is for you a very good introduction to Lambert function.

Sooner or later, you will learn that any equation which write or can rewrite as $$A+Bx+C\log(D+Ex)=0$$ shows explicit solution's in term of Lambert function.

Consider the more general case of $$a x e^x +b e^x=c$$ and apply the steps described in the Wikipedia page and get $$x=W\left(\frac{c }{a}e^{\frac{b}{a}}\right)-\frac{b}{a}$$ Otherwise, only numerical methods will do the job.

Answer 2

So $$(x+1)e^{x+1}=e^5 $$ which makes $$x=W(e^5)-1.$$

Answer 3

One might be interested in approximating the solution:

$$y=x+1$$

$$ye^y=e^5$$

$$e^y=e^5/y$$

$$y=5-\ln(y)$$

$$y_{n+1}=5-\ln(y_n)$$

Apply fixed point iteration with $y_0=5$ to get

$$y_0=5\\y_1=5-\ln(5)\\y_2=5-\ln(5-\ln(5))\\y_3=5-\ln(5-\ln(5-\ln(5)))\\y_4=5-\ln(5-\ln(5-\ln(5-\ln(5))))$$

Which approaches the solution. $y_{10}\approx3.693443808$, and the actual solution is $y\approx3.693441359$, so an error of about $1.5\times10^{-6}$.