NIMCET 2017 — Mathematics PYQ

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Question

NIMCET 2017 — Mathematics PYQ

NIMCET | Mathematics | 2017

The solution of $(e^{x} + 1) y d y = (y + 1) e^{x} d x$ is

Choose the correct answer:

Explanation

Concept:
$lo g x = y \Rightarrow x = e^{y}$

Calculation:

Given: $(e^{x} + 1) y d y = (y + 1) e^{x} d x$

$\Rightarrow \frac{y d y}{y + 1} = \frac{e ^{x}}{( e ^{x} + 1 )} d x$

$\Rightarrow \frac{( y + 1 - 1 ) d y}{y + 1} = \frac{e ^{x}}{( e ^{x} + 1 )} d x$

$\Rightarrow d y - \frac{d y}{y + 1} = \frac{e ^{x}}{( e ^{x} + 1 )} d x$

Integrating both sides:

$\Rightarrow \int d y - \int \frac{d y}{y + 1} = \int \frac{e ^{x}}{e ^{x} + 1} d x$

$\Rightarrow y - lo g (y + 1) = lo g (e^{x} + 1) + c$

$\Rightarrow e^{y} = c (e^{x} + 1) (y + 1)$

Integrating both sides, we get

$\Rightarrow y - lo g ∣ y + 1∣ = lo g (e^{x} + 1) + lo g c$

$\Rightarrow y = lo g ∣ (y + 1) (e^{x} + 1) ∣ + lo g c$

$\Rightarrow c (y + 1) (e^{x} + 1) = e^{y}$

$∴ e^{y} = c (e^{x} + 1) (y + 1)$

Hence, option (1) is correct.