Remark about $E(t)$

$E(t)$ does not measure energy, energy per mass unit.

Starting with an ODE coming directly from Newton's 2nd law $F=ma$ with a spring force $F=-ku$ and $ma=mu^{\prime\prime}$ ($a$: acceleration, $u$: displacement), we have $$mu^{\prime\prime} + ku = 0$$ Integrating this equation gives a physical energy balance: $$E(t) = \underbrace{{\half}mv^2}_{\hbox{kinetic energy} } + \underbrace{{\half}ku^2}_{\hbox{potential energy}} = E(0),\quad v=u^{\prime}$$ Note: the balance is not valid if we add other terms to the ODE.