Question:
If $y=y(x)$ is the solution of the differential
equation, $e^{y}\left(\frac{d y}{d x}-1\right)=e^{x}$ such that $y(0)=0$, then $\mathrm{y}(1)$ is equal to :
Correct Option: , 4
Solution:
$e^{y} \frac{d y}{d x}-e^{y}=e^{x}$, Let $e^{y}=t$
$\Rightarrow \mathrm{e}^{\mathrm{y}} \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{dt}}{\mathrm{dx}}$
$\frac{d t}{d x}-t=e^{x}$
I.F. $=e^{\int-d x}=e^{-x}$
$t e^{-x}=x+c \Rightarrow e^{y-x}=x+c$
$y(0)=0 \Rightarrow c=1$
$e^{y-x}=x+1 \Rightarrow y(1)=1+\log _{e} 2$