What is the answer to xe^{-x}=-3(xe^{-x})+1 ?

The answer to xe^{-x}=-3(xe^{-x})+1 is x=-\W_{-1}(-1/4),x=-\W_{0}(-1/4)

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xe^{-x}=-3(xe^{-x})+1

x e^{- x} = - 3 (x e^{- x}) + 1

x = - W_{- 1} (- \frac{1}{4}), x = - W_{0} (- \frac{1}{4})

Solution steps

x e^{- x} = - 3 (x e^{- x}) + 1

Prepare

x e^{- x} = - 3 (x e^{- x}) + 1

for Lambert form:

4 e^{- x} x = 1

Rewrite the equation with

- x = u

and

x = - u

4 e^{u} (- u) = 1

Rewrite

4 e^{u} (- u) = 1

in Lambert form:

e^{u} u = - \frac{1}{4}

Solve

e^{u} u = - \frac{1}{4} : u = W_{- 1} (- \frac{1}{4}), u = W_{0} (- \frac{1}{4})

u = W_{- 1} (- \frac{1}{4}), u = W_{0} (- \frac{1}{4})

Substitute back

u = - x,

solve for

x

Solve

- x = W_{- 1} (- \frac{1}{4}) : x = - W_{- 1} (- \frac{1}{4})

Solve

- x = W_{0} (- \frac{1}{4}) : x = - W_{0} (- \frac{1}{4})

x = - W_{- 1} (- \frac{1}{4}), x = - W_{0} (- \frac{1}{4})

(6) (8^{x}) = 2 e^{3 x}

A (x) = 500 (1.03)^{x}

4 9^{x + 3} = 7^{x^{2} - 3 x + 8}

8^{x} = 360000000

3^{x + 1} + 3^{x - 1} = 90

What is the answer to xe^{-x}=-3(xe^{-x})+1 ?
The answer to xe^{-x}=-3(xe^{-x})+1 is x=-\W_{-1}(-1/4),x=-\W_{0}(-1/4)