The equation of the curve whose slope at any point is equal to y +...

We have, $\dfrac{dy}{dx} = y + 2x$

$⇒ \dfrac{dy}{dx} - y = 2x$

Now, $IF = e^{- ∫ 1dx} = e^{- x}$

$y \cdot e^{- x} = ∫ 2xe^{- x}dx + k, k$ be the constant of integration

$= 2\left[x ∫ e^{- x}dx - ∫ 1 \cdot \left(- e^{- x}\right)dx\right] + k$ [using integration by parts]

$⇒ y \cdot e^{- x} = - 2xe^{- x} - 2e^{- x} + k... \left(i\right)$

As curve (i) passes through \left(0, 0\right)$</p><p>$∴ 0 = 0 - 2 + k$</p><p>$⇒ k = 2$</p><p>Thus, the curve is</p><p>$ye^{- x} = - 2xe^{- x} - 2e^{- x} + 2$</p><p>$∴ y = 2 \left(e^{x} - x - 1\right)

Hence, option (b) is correct.