Explanation

(b)
$y = a\cos{(\log{x) + b\sin{(\log{x)}}}}$
On differentiating w.r.t. $x$, we get
$y^{'} = \frac{- a\sin{(\log{x)}}}{x} + \frac{b\cos{(\log{x)}}}{x}$
$\Rightarrow xy^{'} = - a\sin{(\log{x) + b\cos{(\log{x)}}}}$
Again, on differentiating w.r.t. $x$, we get
$xy^{''} + y^{'} = - a\cos{(\log{x)\frac{1}{x} - b\sin{(\log{x)\frac{1}{x}}}}}$
$\Rightarrow x^{2}y^{''} + y^{'}x = - \lbrack a\cos{(\log{x) + b\sin{(\log x}}})\rbrack$
$\Rightarrow x^{2}y^{''} + xy^{'} = - y$