If $y(x)$ is a solution of the equation $$4xy’’+2y’+y=0, y(0)=1$$ Then $y’’(0)$ is equal to
$1$. $1/24$.
$2$. $1/12$.
$3$. $1/6$.
$4$. $1/2$.
Actually it was $3$ minutes question in exam so I have to take less time . I was unable to solve it . Firstly I think it can be reduced to Cauchy Euler equation which was not possible by multiple of $x$. Secondly I think reduction of order is possible then one solution is neither given nor easy to find . Power series solution method is not in my syllabus. How to solve it in less time . Please help. Thank you .
Differentiate the DE and note that $y'(0)=-1/2$
$$4xy’’+2y’+y=0$$ $$4xy’'’+6y''+y'=0$$
$\implies 6y''(0)=-y'(0)$ $\implies y''(0)=1/12$