Explanation

(b)
$\frac{dy}{dx} = y + 2x$
$\Rightarrow$ $\frac{dy}{dx} - y = 2x$
$IF = e^{\int_{}^{}{- 1\ dx}} = e^{- x}$
$\therefore$ Solution of the differential equation is
$y.e^{- x} = 2\int_{}^{}{xe^{- x}}\ dx$
$= 2\left( - xe^{- x} - e^{- x} \right) + c$
$\Rightarrow$ $y = 2e^{x}\left( - xe^{- x} - e^{- x} \right) + ce^{x}$
$\Rightarrow$ $y = - 2x - 2 + ce^{x}$
For $c = 2$
We get $y = 2(e^{x} - x - 1)$