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

$$\mathrm{Find}\mathrm{f}\left(\mathrm{x}\right)={\underset{0}{\int }}^{\mathrm{x}}\frac{\mathrm{dt}}{\mathrm{t}+{\mathrm{e}}^{\mathrm{f}\left(\mathrm{t}\right)}}$$

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

$$f\left(x\right)={\int }_0^x\frac{dt}{t+e^{f\left(t\right)}}$$$$f^{\prime }\left(x\right)=\frac{1}{x+e^{f\left(x\right)}}$$$$\frac{dy}{dx}=\frac{1}{x+e^y}$$$$\frac{dx}{dy}=x+e^y$$$$\frac{dx}{dy}-x=e^y\leftarrow D.E.oftype{y}^{\prime }+p\left(x\right)y=q\left(x\right)$$$$\Rightarrow x=\frac{\int e^{-y}{e}^{y}dy+C}{e^{-y}}=(y+C)e^y$$$$\Rightarrow {xe}^C=(y+C)e^{y+C}$$$$\Rightarrow y+C=W\left({xe}^C\right)\leftarrow LambertWfunction$$$$or$$$$y=f\left(x\right)=W\left(kx\right)-\ln k$$$$f\left(0\right)=W\left(0\right)-\ln k=0\Rightarrow \ln k=0\Rightarrow k=1$$$$\Rightarrow f\left(x\right)=W\left(x\right)$$

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