# What is a particular solution to the differential equation dy/dx=e^(x−y) with y(0)=2?

What is a particular solution to the differential equation $\frac{dy}{dx}={e}^{x-y}$ with y(0)=2?
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tiepidolu
this is separable
$\frac{dy}{dx}={e}^{x-y}={e}^{x}{e}^{-y}$
${e}^{y}\frac{dy}{dx}={e}^{x}$

${e}^{y}={e}^{x}+C$
$y\left(0\right)=2⇒{e}^{2}=1+C⇒C={e}^{2}-1$
${e}^{y}={e}^{x}+{e}^{2}-1$
$y=\mathrm{ln}\left({e}^{x}+{e}^{2}-1\right)$