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Step 3: Approximate derivative(s) by finite difference approximation(s). Very common (standard!) formula for \( u^{\prime\prime} \): $$ u^{\prime\prime}(t_n) \approx \frac{u^{n+1}-2u^n + u^{n-1}}{\Delta t^2} $$
Use this discrete initial condition together with the ODE at \( t=0 \) to eliminate \( u^{-1} \): $$ \frac{u^{n+1}-2u^n + u^{n-1}}{\Delta t^2} = -\omega^2 u^n $$
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