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\( E(t) \) 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.
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