In my question here I want to know from the following ODE which ODE have oscillatory solutions \begin{equation} y''(x)+(\sin^2 x+1)y(x)=0\tag{1} \end{equation} \begin{equation} y''(x)-x^2y(x)=0\tag{2} \end{equation} \begin{equation} y''(x)+\dfrac{1}{x}y(x)=0\tag{3} \end{equation}
Attempt:I have used the classical theorem of Sturm which assert for solutions of: $y''+a(x) y =0,x>0 $ , If $a(x)\geq a_0>0\implies \text{Oscillation}$ and $a(x)\leq a_0<0\implies \text{NonOscillation}$, From this theorem We have only solution of Equation $(1)$ which oscillatory solutions, because we have :$\sin^2 x+1 \geq 1 > 0$ and it is a Mathieu differential equation which it is Oscillatory, but its seems to me that is not enough because this theorem is not applied for $a(x)=\dfrac{1}{x}$ because $a(0)$ is undeterminted , Another Idea I have used the folowing definition:
Definition: We say that solutions of $y''+a(x) y =0,x>0 $ are oscillatory if its solutions have infinity many zeros this necessite to compute this integral: $N(x)=\dfrac{1}{\pi}\displaystyle \int_{a}^{x}\sqrt{a(s)}ds$ where $N(x)$ is the number of zeros of solution of each ODE,$x\to \infty $, From this definition seems solutions of all above equation are oscillatory because after computation of integrals we have divergence to $\infty $ which means there are infinity many zeros of solutions of all above ODE, My question here is : Which above equations have oscillatory solutions ?