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]
