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How do you reduce a 2nd-order DE d²x/dt² = g(t, x, dx/dt) to first-order equations?
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All Flashcards in Topic 5.18
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5.18.18 cards
How do you reduce a 2nd-order DE d²x/dt² = g(t, x, dx/dt) to first-order equations?
Let v = dx/dt. Then dx/dt = v and dv/dt = g(t, x, v) — a coupled first-order system.
Write the Euler update for a coupled 2nd-order system.
x_(n+1) = x_n + h·v_n; v_(n+1) = v_n + h·g(t_n, x_n, v_n); t_(n+1) = t_n + h.
In one Euler step of a coupled system, which values feed the rates?
All rates use the OLD row's values; update x and v together, then advance t — never reuse a freshly-updated value within the same step.
How many Euler steps reach a target time t_end?
steps = (t_end − t_0) ÷ h. Always re-check this count — the off-by-one error is the classic trap.
How does the step length h affect Euler's accuracy?
A smaller h is more accurate (shorter tangent steps stray less from the true curve) but needs more steps.
Reduce d²x/dt² = −4x with v = dx/dt.
dx/dt = v and dv/dt = −4x (a spring with k = 4).
How do you find the percentage error of an Euler estimate?
|estimate − exact| ÷ |exact| × 100%.
Which AI exam paper most features 2nd-order Euler / coupled systems?
Paper 3 — the extended modelling investigations (often with phase portraits and eigenvalue DE solutions). A GDC is allowed throughout.
Topic 5.18 study notes
Full notes & explanations for Euler's method (2nd order) (HL only)
Math AI exam skills
Paper structures, command terms & tips
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