Answer
(a) $J'(0)$ = $0$
(b) $J''(0)$ = $-\frac{1}{2}$
Work Step by Step
(a)
$y$ = $J(x)$
$xy''+y'+xy$ = $0$
$xJ''(x)+J'(x)+xJ(x)$ = $0$
if $x$ = $0$
we have
$0+J'(x)+0$ = $0$
$J'(0)$ = $0$
(b)
Differentiating implicit
$xy''+y'+xy$ = $0$
$xy'''+y''+y''+xy'+y$ = $0$
$xy'''+2y''+xy'+y$ = $0$
if $x$ = $0$
$0+2J''(0)+0+1$ = $0$
$J''(0)$ = $-\frac{1}{2}$
