Find-f-x-x-0-dt-t-e-f-t-

Question Number 209246 by Erico last updated on 05/Jul/24

Find f (x) = {\int_{0}}^{x} \frac{dt}{t + e^{f (t)}}

Answered by mr W last updated on 06/Jul/24

f (x) = \int_{0}^{x} \frac{d t}{t + e^{f (t)}}

f^{'} (x) = \frac{1}{x + e^{f (x)}}

\frac{d y}{d x} = \frac{1}{x + e^{y}}

\frac{d x}{d y} = x + e^{y}

\frac{d x}{d y} - x = e^{y} \leftarrow D . E . o f t y p e y^{'} + p (x) y = q (x)

\Rightarrow x = \frac{\int e^{- y} e^{y} d y + C}{e^{- y}} = (y + C) e^{y}

\Rightarrow {x e}^{C} = (y + C) e^{y + C}

\Rightarrow y + C = W ({x e}^{C}) \leftarrow L a m b e r t W f u n c t i o n

o r

y = f (x) = W (k x) - \ln k

f (0) = W (0) - \ln k = 0 \Rightarrow \ln k = 0 \Rightarrow k = 1

\Rightarrow f (x) = W (x)

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