correction for last part:

if $f(y)=y$ , then the differential equation is $y'' + y =0$, meaning that

$y=A cosx +B Sinx$ and $z=-y'= - A sinx +B cosx$

are the rotate axes.

$pA^2+qAB+rB^2=1$

$p,q,r$ depends on $x$

choose $x$ such that $q=0$

$pA^2+rB^2=1$

what can i do after that?