Question

If $y=y(x)$ is the solution of the differential equation, $e^y\left(\frac{dy}{dx}-1\right)=e^x$ such that $y\left(0\right)=0$, then $y\left(1\right)$ is equal to :

• A. $lo{g}_{e}2$
• B. $2e$
• C. $2+lo{g}_{e}2$
• D. $1+lo{g}_{e}2$

Answer 1

Answer: (D)

$e^y\left(\frac{dy}{dx}-1\right)=e^y=e^x$ , Let $e^y=t$
$\Rightarrow e^y\frac{dy}{dx}=\frac{dt}{dx}$
$\frac{dt}{dx}-t=e^x$
$I.F=e^{\int -dx}=e^{-x}$
$t{e}^{-x}=x+c\Rightarrow e^{y-x}=x+c$
$y\left(0\right)=0\Rightarrow c=1$
$e^{y-x}=x+1\Rightarrow y\left(1\right)=1+lo{g}_e^2$