Explanation

(a)
Given $y + x^{2} = \frac{dy}{dx}\ \Rightarrow \ \frac{dy}{dx} - y = x^{2}$
This is the linear differential equation of the form
$\frac{dy}{dx} + Py = Q$
$\Rightarrow P = - 1,\ Q = x^{2}$
$\therefore IF = e^{\int_{}^{}P\ dx} = e^{\int_{}^{}{- 1\ dx}} = e^{- x}$
Hence, required solution is
$ye^{- x} = \int_{}^{}{x^{2}e^{- x}\ dx}$
$y \bullet e^{- x} = - x^{2}e^{- x} - 2xe^{- x} - 2e^{- x} + c$
$\Rightarrow y + x^{2} + 2x + 2 = ce^{x}$