Let $$y''-q(x)y=0$$be a differential equation with initial conditions on $0\leq x<\infty,$ as $y(0)=1,y'(0)=1$ where $q(x)$ is a positive monotonically increasing continuous function. Then which of the following are true?
$y(x)\rightarrow\infty$ as $x\rightarrow\infty$.
$y'(x)\rightarrow\infty$ as $x\rightarrow\infty$.
$y(x)$ has finitely many zeros in $[0,\infty)$.
$y(x)$ has infinitely many zeros in $[0,\infty)$.
Please don't mind, actually I am new in differential equation. I only know that by Picard's theorem the above differential equation has the unique solution, but I don't know what is the solution as I tried by direct hit and trial method. According to me the solution of above differential equation will be some thing in exponential form, so according to me its answer will be $a$, $b$, $c$. But I don't know the exact method. Please help me to solve the above problem. Thanks in advance.