The general solution of dydx=ex-y(ex-ey) is

dydx=exe−y(ex−ey)=e2xe−y−exdydx+ex=e2xey⇒ ey dydx+eyex=e2xLet ey=veydydx=dvdx⇒ dvdx+exv=e2xI.F =e∫exdx=eexSolution isv(I.F)=∫e2xeexdxex=t⇒ exdx=dt⇒ eyeex=∫t et dt⇒ t et−et+c⇒ exeex−eex+c⇒ ey= (ex−1)+ce−ex

The general solution of dydx=ex-y(ex-ey) is

dydx=exe−y(ex−ey)=e2xe−y−exdydx+ex=e2xey⇒ ey dydx+eyex=e2xLet ey=veydydx=dvdx⇒ dvdx+exv=e2xI.F =e∫exdx=eexSolution isv(I.F)=∫e2xeexdxex=t⇒ exdx=dt⇒ eyeex=∫t et dt⇒ t et−et+c⇒ exeex−eex+c⇒ ey= (ex−1)+ce−ex

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The general solution of $\frac{dy}{dx} = e^{x - y} (e^{x} - e^{y})$ is

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$e^{y} = e^{x} + 1 + c . e^{- ex}$

$e^{y} = e^{x} - 2 - c . e^{- ex}$

$e^{y} = e^{x} - 1 - c . e^{- x}$

$e^{y} = e^{x} - 1 + c . e^{- ex}$

answer is A.

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Detailed Solution

$\begin{array}{l} \frac{d y}{d x} = e^{x} e^{- y} (e^{x} - e^{y}) \\ = e^{2 x} e^{- y} - e^{x} \\ \frac{d y}{d x} + e^{x} = \frac{e^{2 x}}{e^{y}} \\ \Rightarrow e^{y} \frac{d y}{d x} + e^{y} e^{x} = e^{2 x} \\ L e t e^{y} = v \\ e^{y} \frac{d y}{d x} = \frac{d v}{d x} \\ \Rightarrow \frac{d v}{d x} + e^{x} v = e^{2 x} \end{array}$

$\begin{array}{l} I . F = e^{\int e^{x} d x} = e^{e^{x}} \\ S o l u t i o n i s \\ v (I . F) = \int e^{2 x} e^{e^{x}} d x \\ e^{x} = t \Rightarrow e^{x} d x = d t \\ \Rightarrow e^{y} e^{e^{x}} = \int t e^{t} d t \\ \Rightarrow t e^{t} - e^{t} + c \\ \Rightarrow e^{x} e^{e^{x}} - e^{e^{x}} + c \\ \Rightarrow e^{y} = (e^{x} - 1) + c e^{- e^{x}} \end{array}$