# Thread: solving e^x=5x+1

solving
e^x=5x+1
thanks

2. ## Re: solving e^x=5x+1

there is no exact solution for power-algebric equation

3. ## Re: solving e^x=5x+1

Originally Posted by narjsingh
solving e^x=5x+1
one solution determined by "observation" is ...

$\displaystyle x = 0$

the other solution can be approximated using technology ...

$\displaystyle x \approx 2.6603991$

4. ## Re: solving e^x=5x+1

The non-evident root cannot be expressed analyticaly with the combination of a finite number of usual functions.
It can be erxpressed with infinite series (complicated, not useful in practice)
Usally, this kind of equation is solved thanks to numerical computation.
The root can be expressed on a closed form thanks to the Lambert W function :
x = -W(X) where X = -(exp(-1/5)-1)/5

5. ## Re: solving e^x=5x+1

Originally Posted by skeeter
one solution determined by "observation" is ...

$\displaystyle x = 0$

the other solution can be approximated using technology ...

$\displaystyle x \approx 2.6603991$
In this case, It is ok, but in general it is diffecult to solve such equation directly
for example : solve : e^x=x+2

6. ## Re: solving e^x=5x+1

Originally Posted by Kyood
In this case, It is ok, but in general it is diffecult to solve such equation directly
for example : solve : e^x=x+2
solve exp(x)=x+2 for x - Wolfram|Alpha

7. ## Re: solving e^x=5x+1

Originally Posted by Kyood
In this case, It is ok, but in general it is diffecult to solve such equation directly
for example : solve : e^x=x+2
I'm pleased that my solution meets your approval ... in "this case".

8. ## Re: solving e^x=5x+1

Very nice, skeeter,
you used a special function W(x) which consedered as an approximate solutuion.

9. ## Re: solving e^x=5x+1

Originally Posted by Kyood
Very nice, skeeter,
you used a special function W(x) which consedered as an approximate solutuion.
no ... I just solved $\displaystyle e^x - 5x - 1 = 0$ with a calculator.

10. ## Re: solving e^x=5x+1

$\displaystyle \int_{0}^{\pi}\frac{x^{4}\left(1-x\right)^{4}}{1+x^{2}}dx =\frac{22}{7}-\pi$

11. ## Re: solving e^x=5x+1

Originally Posted by divyaprashanth07
$\displaystyle \int_{0}^{\pi}\frac{x^{4}\left(1-x\right)^{4}}{1+x^{2}}dx =\frac{22}{7}-\pi$
... what does this incorrectly evaluated definite integral have to do with this thread?