$$ y'' + \lambda y = 0, \quad y'(0) = 0 \quad |y(x)| < \infty \quad \text{for all } y \text{ in } (0, \infty),$$
I have tried numerous Sturm Liouville Boundary Value problems, but never done problems involving boundary conditions where there are inequalities and infinities. Is this a regular SLBVP or does it classify as a singular SLBVP.
How should I solve this?
The condition $|y(x)|<\infty$ just means that the function is real (doesn't take on infinite values).
Thus, you can solve this as a normal equation, finding that the general solution is (if $\lambda>0$, other cases are left to the reader)$$y(x)=c_1\sin(\sqrt{\lambda}x)+c_2\cos(\sqrt{\lambda}x)$$
By forcing $y'(0)=0$ one finds $c_1=0$ and the solution is $c_2\cos(\sqrt{\lambda}x)$.