Let $m,k>0$, and $A,B$ be real. Then the differential equation

$$m{{d^{2}f}\over{dx^{2}}}=kf(x)$$

with initial conditions

$$f(0)=A,\quad f^{\prime}(0)=B$$

has solution

$$f(x)=A\cosh\beta x+{{B}\over{\beta}}\sinh\beta x$$

where $\beta=\sqrt{k/m}$.

This solution is not periodic, and does not oscillate. Usually $f(x)$ will diverge to $\infty$ or $-\infty$ as $x\rightarrow\infty$.