What is the answer to 2=0.1x+e^{0.25x} ?

The answer to 2=0.1x+e^{0.25x} is x=-(0.1\W_{0}(\frac{0.25e^5)/(0.1))-0.5}{0.025}

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2=0.1x+e^{0.25x}

2 = 0.1 x + e^{0.25 x}

x = - \frac{0.1 W _{0} ( \frac{0.25 e ^{5}}{0.1} ) - 0.5}{0.025}

Decimal Notation

x = 2.28688 \dots

Solution steps

2 = 0.1 x + e^{0.25 x}

Prepare

2 = 0.1 x + e^{0.25 x}

for Lambert form:

1 = (2 - 0.1 x) e^{- 0.25 x}

Rewrite the equation with

- 0.25 x + \frac{0.5}{0.1} = u

and

x = - \frac{u - 5}{0.25}

1 = (2 - 0.1 (- \frac{u - 5}{0.25})) e^{- 0.25 (- \frac{u - 5}{0.25})}

Simplify

1 = e^{u - 5} (2 + \frac{0.1 ( u - 5 )}{0.25})

Rewrite

1 = e^{u - 5} (2 + \frac{0.1 ( u - 5 )}{0.25})

in Lambert form:

e^{u} u = \frac{0.25 e ^{5}}{0.1}

Solve

e^{u} u = \frac{0.25 e ^{5}}{0.1} : u = W_{0} (\frac{0.25 e ^{5}}{0.1})

u = W_{0} (\frac{0.25 e ^{5}}{0.1})

Substitute back

u = - 0.25 x + \frac{0.5}{0.1},

solve for

x

Solve

- 0.25 x + \frac{0.5}{0.1} = W_{0} (\frac{0.25 e ^{5}}{0.1}) : x = - \frac{0.1 W _{0} ( \frac{0.25 e ^{5}}{0.1} ) - 0.5}{0.025}

x = - \frac{0.1 W _{0} ( \frac{0.25 e ^{5}}{0.1} ) - 0.5}{0.025}

y, x y = e^{x + y}

9^{x^{2}} = 3^{x + 3}

3^{2 x} = 7

2^{3 x} - 2^{2 x} = 2^{x + 1}

9^{2 x - 1} = 3^{6 x}

What is the answer to 2=0.1x+e^{0.25x} ?
The answer to 2=0.1x+e^{0.25x} is x=-(0.1\W_{0}(\frac{0.25e^5)/(0.1))-0.5}{0.025}