What is the answer to solvefor t,f=ae^{-at}u(t) ?

The answer to solvefor t,f=ae^{-at}u(t) is t=-(\W(-f/u))/a

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solvefor t,f=ae^{-at}u(t)

solve for

t, f = a e^{- a t} u (t)

t = - \frac{W ( - \frac{f}{u} )}{a}

Solution steps

f = a e^{- a t} u (t)

Rewrite the equation with

- a t = v

and

t = - \frac{v}{a}

f = a e^{v} u (- \frac{v}{a})

Rewrite

f = a e^{v} u (- \frac{v}{a})

in Lambert form:

e^{v} v = - \frac{f}{u}

Solve

e^{v} v = - \frac{f}{u} : v = W (- \frac{f}{u})

v = W (- \frac{f}{u})

Substitute back

v = - a t,

solve for

t

Solve

- a t = W (- \frac{f}{u}) : t = - \frac{W ( - \frac{f}{u} )}{a}

t = - \frac{W ( - \frac{f}{u} )}{a}

2^{4 x} - 5 \cdot 2^{2 x} + 4 = 0

9 \cdot 2 7^{x} = 27

1.7 = 240 (0.5)^{\frac{x}{19}}

n, (z^{2})^{n} = z^{10}

3^{x^{2} - 12} = 9^{2 x}

What is the answer to solvefor t,f=ae^{-at}u(t) ?
The answer to solvefor t,f=ae^{-at}u(t) is t=-(\W(-f/u))/a