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Q. 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 :

JEE MainJEE Main 2020Differential Equations

Solution:

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