The equation of one of the curves whose slope at any point is equal to $y + 2 x$ is

y = 2 (e^{x} + x - 1)

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y = 2 (e^{x} - x - 1)

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y = 2 (e^{x} - x + 1)

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y = 2 (e^{x} + x + 1)

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Solution

The correct option is C $y = 2 (e^{x} - x - 1)$
Given, $\frac{d y}{d x} = y + 2 x$
Put $y + 2 x = z$
$\Rightarrow \frac{d y}{d x} + 2 = \frac{d z}{d x}$
$\Rightarrow \frac{d y}{d x} = \frac{d z}{d x} - 2 \dots (i i)$
From Eqs. (i) and (ii)
$\frac{d z}{d x} - 2 = z$
$\Rightarrow \int \frac{d z}{. z + 2} = \int d x$
$\Rightarrow l o g (z + 2) = x + c$
$\Rightarrow l o g (y + 2 x + 2) = x + c$
$\Rightarrow y + 2 x + 2 = e^{x + c}$
$\Rightarrow y + 2 x + 2 = e^{x} . e^{c}$
$\Rightarrow y = 2 [e^{x} - x - 1]$

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