In addition to the periodicity test, the total energy has also to be conserved.

Suppose we choose the initial conditions $$ x(t=0)=1\hspace{0.1cm} \mathrm{m}\hspace{1cm} v(t=0)=0\hspace{0.1cm}\mathrm{m/s}, $$ meaning that block is at rest at \( t=0 \) but with a potential energy $$ E_0=\frac{1}{2}kx(t=0)^2=\frac{1}{2}k. $$ The total energy at any time \( t \) has however to be conserved, meaning that our solution has to fulfil the condition $$ E_0=\frac{1}{2}kx(t)^2+\frac{1}{2}mv(t)^2. $$