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I got an equation $x^{a}e^{bx}=c$, which I would like to find the solution of $x$. Could someone please give me some hints? Thanks a lot!

Gary
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Joey
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    Might wanna look at this. – Anthony Jul 26 '22 at 06:49
  • Did you really get this equation in all of its generality (i.e. for arbitrary $a, b, c$)? Or, have you possibly got concrete $a,b,c$ but you think that knowing how to solve it in general will help you with your particular problem? If the latter, please let us know. Although not solvable in general (except by using special functions), this equation may be solvable in elementary ways in some particular cases. –  Jul 26 '22 at 08:09

1 Answers1

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In general, there is no meaningful way without the Lambert $W$ function; it is defined as the inverse function to $f(x) = xe^x$ (over appropriate domains). Hence

$$W(xe^x) = W(x) e^{W(x)} = x$$

Take the $a$th root of each side:

$$x e^{(b/a) x } = c^{1/a}$$

Multiply by $b/a$:

$$\frac b a x \cdot e^{(b/a)x} = \frac b a c^{1/a}$$

Apply the $W$ function:

$$\frac b a x = W \left( \frac b a c^{1/a} \right)$$

Multiply by $a/b$:

$$x = \frac a b W \left( \frac b a c^{1/a} \right)$$

PrincessEev
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    This question seems not to meet the standards for the site. Instead of answering it, it would be better to look for a good duplicate target, or help the user by posting comments suggesting improvements. Please also read the meta announcement regarding quality standards. – Sourav Ghosh Jul 26 '22 at 07:14