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Question Number 88003 by Ar Brandon last updated on 07/Apr/20

$$Determineallfunctionsf[0,1]\to \mathrm{\Omega }$$$$suchthat\mathrm{\forall }x\in [0,1]f{}^{\prime }\left(x\right)+f\left(x\right)=f\left(0\right)+f\left(1\right)$$

Commented by mr W last updated on 07/Apr/20

$$f\left(0\right)+f\left(1\right)=k$$$$f^{\prime }\left(x\right)+f\left(x\right)=k$$$$\frac{dy}{y-k}=-dx$$$$\ln \mid y-k\mid =-x+C$$$$y=k+{ce}^{-x}$$$$f\left(0\right)=k+c$$$$f\left(1\right)=k+\frac{c}{e}$$$$f\left(0\right)+f\left(1\right)=k+c+k+\frac{c}{e}=k$$$$k=-c(1+\frac{1}{e})$$$$\Rightarrow y=f\left(x\right)=c(e^{-x}-1-\frac{1}{e})$$

Commented by Ar Brandon last updated on 07/Apr/20

$$Thanks$$

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