every non trivial solution of the equation $y''+(\sinh x)y=0$ has at most one zero

Question

Show that every non trivial solution of the equation

$$y''+(\sinh x)y=0$$

has at most one zero in $(-\infty, 0)$ and inifintly many zeros in $(0,\infty)$

how can we use here sturm comparison theorem

Answer 1

The important part is that $\sinh x>0$ for $x>0$ and $\sinh x<0$ for $x<0$. Everything else follows from that.

For the negative half axis if $-a$ is the smallest negative root of a solution $y$, compare to the solutions of $u''-\sinh(c)u=0$ for $x\le -a$ and some $c\in (0,a)$.

For the positive half axis, compare with $u''+\sinh(1)u=0$ for $x\ge 1$ to find infinitely many zeros.