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. $log_e\,2$
• B. $2e$
• C. $2 + log_e2$
• D. $1 + log_e2$

Answer 1

Answer: (D)

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