Tell me more ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.
up vote 1 down vote favorite

How to solve for the equation $ax \exp(bx)=c$? It is known that $x\geq 0$.

share|improve this question

1 Answer

up vote 2 down vote

Solving this equation requires the Lambert W-function, which when applied to c gives the solution for a=b=1. Barring trivial cases, I don't think there's any easier way to derive it.

share|improve this answer
2  
In general, we can rewrite the equation as $(bx) \exp(bx) = \frac{bc}{a} = d$ (say). The solution for this is $bx = W(d)$, or $x = \frac{1}{b}W(d)$. – Srivatsan Nov 23 '11 at 22:55
Srivatsan's comment ought to be the answer, methinks. – J. M. Nov 24 '11 at 8:33
Thanks, it is helpful. And what about more general equation ax*exp(bx+p)=c? And how to find a solution for W(d), if d>=0 and is supposed to be a real number? – Sergei Nov 25 '11 at 0:14
ax*exp(bx)*exp(p)=c, then bx*exp(bx)=b*(c/a)*exp(-p)=d. Am I right? Let d = 5. How to find the value of W(d)? Taylor series expansion around 0? – Sergei Nov 25 '11 at 0:33
@Sergei: Why not ask a new question? – J. M. Dec 4 '11 at 4:54

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.