🗝️ Key Takeaways ...

• Analytic solutions are exact; numerical solutions are approximate.
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• Numerical solutions are essential when analytic solutions are too complicated or don’t exist.
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• Euler’s method approximates solutions by following the slope given by the differential equation, taking small steps from one point to the next.
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• Euler’s one-step task is “move from where you are in the direction of the slope.”
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• Choosing a step size \(h\) sets the “run” = \(h\) and “rise” = slope \(\times\ h\text{.}\)
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• Euler’s method is based on the movement rule

  \begin{equation*} (t_{\text{new}}, y_{\text{new}}) = (t_{\text{cur}} + h,\ y_{\text{cur}} + h \times \text{slope})\text{,} \end{equation*}

  so each step is given by: \(y_{k+1} = y_k + h f(t_k, y_k)\text{.}\)

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• Smaller step sizes usually mean better accuracy — but more steps.
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• Holding the starting slope fixed across a step costs about \(\frac12 y'' h^2\text{,}\) but covering a fixed interval takes \(N = (t_N - t_0)/h\) steps, so one power of \(h\) is spent on the extra steps. Euler is a first-order method: its global error is proportional to \(h\text{,}\) so halving \(h\) roughly halves the final error rather than quartering it.

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• Improved Euler (Heun’s method) predicts with an ordinary Euler step, reads a second slope where the predictor lands, and re-steps using the average of the two:

  \begin{equation*} y_{k+1} = y_k + \frac{h}{2}\Big[\, f(t_k, y_k) + f\big(t_{k+1},\ y_k + h f(t_k, y_k)\big) \Big]. \end{equation*}

  It is second order — halving \(h\) quarters the error — which is worth far more than the doubled cost per step.

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• Accuracy is bought more cheaply by using a better method than by using a smaller step size.
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