en

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

3=x+3e^{-x}

Solution

3 = x + 3 e^{- x}

Solution

x = 0, x = W_{0} (- \frac{3}{e ^{3}}) + 3

+1

Decimal Notation

x = 0, x = 2.82143 \dots

Solution steps

3 = x + 3 e^{- x}

Prepare

3 = x + 3 e^{- x}

for Lambert form:

3 = (3 - x) e^{x}

Rewrite the equation with

x - 3 = u

and

x = u + 3

3 = (3 - (u + 3)) e^{(u + 3)}

Simplify

3 = - e^{u + 3} u

Rewrite

3 = - e^{u + 3} u

in Lambert form:

e^{u} u = - \frac{3}{e ^{3}}

Solve

e^{u} u = - \frac{3}{e ^{3}} : u = - 3, u = W_{0} (- \frac{3}{e ^{3}})

u = - 3, u = W_{0} (- \frac{3}{e ^{3}})

Substitute back

u = x - 3,

solve for

x

Solve

x - 3 = - 3 : x = 0

Solve

x - 3 = W_{0} (- \frac{3}{e ^{3}}) : x = W_{0} (- \frac{3}{e ^{3}}) + 3

x = 0, x = W_{0} (- \frac{3}{e ^{3}}) + 3

Popular Examples

7^{x} = 7^{3}

4^{7x-2}=17^{x+1}

4^{7 x - 2} = 1 7^{x + 1}

6^{x} = 3 6^{3}

5^{2x+3}=6^{x-1}

5^{2 x + 3} = 6^{x - 1}

3^{x} = 3^{- x}

Frequently Asked Questions (FAQ)

What is the answer to 3=x+3e^{-x} ?
The answer to 3=x+3e^{-x} is x=0,x=\W_{0}(-3/(e^3))+3