What is the answer to e^{0.5x}=5-5x ?

The answer to e^{0.5x}=5-5x is x=-(\W_{0}(\frac{e^{1/2})/(10))-0.5}{0.5}

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e^{0.5x}=5-5x

e^{0.5 x} = 5 - 5 x

x = - \frac{W _{0} ( \frac{e ^{\frac{1}{2}}}{10} ) - 0.5}{0.5}

Solution steps

e^{0.5 x} = 5 - 5 x

Prepare

e^{0.5 x} = 5 - 5 x

for Lambert form:

1 = (5 - 5 x) e^{- 0.5 x}

Rewrite the equation with

- 0.5 x + 0.5 = u

and

x = - 2 u + 1

1 = (5 - 5 (- 2 u + 1)) e^{- 0.5 (- 2 u + 1)}

Simplify

1 = 10 e^{- 0.5 (- 2 u + 1)} u

Rewrite

1 = 10 e^{- 0.5 (- 2 u + 1)} u

in Lambert form:

e^{\frac{1}{2} - 0.5 (- 2 u + 1)} u = \frac{e ^{\frac{1}{2}}}{10}

Solve

e^{\frac{1}{2} - 0.5 (- 2 u + 1)} u = \frac{e ^{\frac{1}{2}}}{10} : u = W_{0} (\frac{e ^{\frac{1}{2}}}{10})

u = W_{0} (\frac{e ^{\frac{1}{2}}}{10})

Substitute back

u = - 0.5 x + 0.5,

solve for

x

Solve

- 0.5 x + 0.5 = W_{0} (\frac{e ^{\frac{1}{2}}}{10}) : x = - \frac{W _{0} ( \frac{e ^{\frac{1}{2}}}{10} ) - 0.5}{0.5}

x = - \frac{W _{0} ( \frac{e ^{\frac{1}{2}}}{10} ) - 0.5}{0.5}

e^{- 2 x} = 5

16 e^{4 x} = 0

2 x e^{- x} - x^{2} e^{- x} = 0

x e^{x} = 2

0 = e^{2 x}

What is the answer to e^{0.5x}=5-5x ?
The answer to e^{0.5x}=5-5x is x=-(\W_{0}(\frac{e^{1/2})/(10))-0.5}{0.5}