Answer :

To solve the equation [tex]\( e^x - x e^{5x - 1} = 0 \)[/tex], follow these steps:

1. Rewrite the equation:

Given equation:
[tex]\[ e^x - x e^{5x - 1} = 0 \][/tex]

2. Isolate one term involving [tex]\( e^x \)[/tex]:

Move [tex]\( x e^{5x - 1} \)[/tex] to the right side of the equation:
[tex]\[ e^x = x e^{5x - 1} \][/tex]

3. Express the equation in a form involving Lambert W function:

Recall that the Lambert W function is defined as the inverse function of [tex]\( z = W(z) e^{W(z)} \)[/tex]. We aim to use this form to our advantage.

First, we simplify the right-hand side:
[tex]\[ e^x = x e^{5x - 1} \][/tex]
Rewrite the exponent on the right-hand side:
[tex]\[ e^x = x e^{5x} e^{-1} \][/tex]
Multiply both sides by [tex]\( e \)[/tex] to simplify:
[tex]\[ e \cdot e^x = x e^{5x} \][/tex]
Simplify the left-hand side:
[tex]\[ e^{x + 1} = x e^{5x} \][/tex]

4. Isolate the exponential term:

Divide both sides by [tex]\( e^{5x} \)[/tex]:
[tex]\[ \frac{e^{x+1}}{e^{5x}} = x \][/tex]
Simplify the left-hand side:
[tex]\[ e^{x + 1 - 5x} = x \][/tex]
Simplify the exponent:
[tex]\[ e^{1 - 4x} = x \][/tex]

5. Introduce the Lambert W function:

To solve for [tex]\( x \)[/tex] using the Lambert W function, set [tex]\( 1 - 4x = -t \)[/tex], so:
[tex]\[ e^{-t} = x \][/tex]
This gives us:
[tex]\[ x e^t = 1 \][/tex]

Replace [tex]\( x \)[/tex] with [tex]\( e^{-t} \)[/tex]:
[tex]\[ e^{-t} e^t = 1 \][/tex]
Simplify:
[tex]\[ e^0 = 1 \][/tex]

Hence:
[tex]\[ x = e^{-t} \][/tex]

6. Express [tex]\( t \)[/tex] in terms of [tex]\( x \)[/tex]:

Recall [tex]\( t = 1 - 4x \)[/tex], so:
[tex]\[ t = 1 - 4x \][/tex]
[tex]\( -t = - (1 - 4x) \)[/tex], thus:
[tex]\[ x = e^{-(1 - 4x)} \][/tex]

Therefore, rewrite it as:
[tex]\[ x = e^{4x - 1} \][/tex]

7. Apply the Lambert W function:

Recall the natural logarithm form:
[tex]\[ 4x - 1 = W(1) = W(e^{4x - 1}) \][/tex]

Solving it for [tex]\( x \)[/tex] results in:
[tex]\[ x = \frac{W(e^4)}{4} = \frac{W(4e)}{4} \][/tex]

Therefore, the solution to the equation [tex]\( e^x - x e^{5x - 1} = 0 \)[/tex] is:

[tex]\[ x = \frac{\text{LambertW}(4e)}{4} \][/tex]