📌 Core idea & why it works
- Conversion: Second-order differential equations are rewritten as coupled first-order systems so numerical methods can be applied systematically.
- Local linearisation: Euler’s method assumes the solution behaves linearly over very small intervals, following tangent lines step-by-step.
- Accuracy: Euler is a first-order method, meaning total numerical error grows proportionally with the chosen step size.
- Stability: Large step sizes may cause oscillations or divergence, revealing limitations of simple numerical schemes.
📱 GDC Tips
- TI-Nspire CX II: Use Lists & Spreadsheet → define columns for t, x, y → enter recursive formulas referencing previous rows.
- TI-Nspire CX II: Use Data & Statistics to plot x against t and visually assess numerical stability and convergence.
- Casio fx-CG50 / CG100: Use TABLE mode → manually define recursion using step size h and verify using graph mode.
- Casio fx-CG50 / CG100: Reduce step size gradually and compare tables to ensure numerical results stabilise.
🧠 Examiner Tip
- Always state the step size h and show at least two explicit Euler updates.
- Clear tables earn method marks even if the final value is inaccurate.
- Comment briefly on expected error behaviour when justifying results.
📌 Euler’s method for d²x/dt² = f(x, dx/dt, t)
- Introduce velocity variable: Let y = dx/dt to transform the equation into dx/dt = y and dy/dt = f(x,y,t).
- Choose step size: Select a constant step length h based on desired accuracy and computational practicality.
- Apply update rules: Advance both position and velocity using current values at each discrete time step.
- Iterate forward: Repeat updates until the final time is reached, recording all intermediate values.
📐 IA Spotlight
- Model damped oscillations, cooling curves, or population recovery using Euler’s method.
- Compare Euler with RK4 to evaluate numerical accuracy and computational efficiency.
- Include convergence tests and error graphs to strengthen mathematical justification.
📌 Practice Questions
MCQ 1
Euler’s method approximates solutions by:
- A. solving differential equations exactly
- B. following tangent lines step-by-step
- C. averaging slopes over intervals
- D. fitting polynomials to data
Answer
B — Euler’s method uses local tangent slopes for approximation.
MCQ 2
Reducing step size h primarily:
- A. increases instability
- B. decreases computational cost
- C. improves accuracy
- D. removes all numerical error
Answer
C — Smaller h reduces truncation error.
MCQ 3
Euler’s method is classified as:
- A. second-order accurate
- B. implicit method
- C. first-order numerical method
- D. exact solver
Answer
C — Euler is first-order accurate.
Short Question 1
Explain why Euler’s method may fail for oscillatory systems.
Answer
Large step sizes may overshoot turning points, causing artificial energy growth or decay and numerical instability.
Short Question 2
State one advantage and one limitation of Euler’s method.
Answer
Advantage: simple and intuitive. Limitation: low accuracy for rapidly changing systems.
Long Question 1
A particle satisfies d²x/dt² = −x with x(0)=1, dx/dt(0)=0.
(a) Rewrite as a first-order system.
(b) Apply one Euler step with h=0.1.
(c) Comment on long-term reliability.
Answer
(a) Let y=dx/dt → dx/dt=y, dy/dt=−x.
(b) x₁=1, y₁=−0.1.
(c) Errors accumulate, causing drift from true oscillatory motion.
Long Question 2
Explain why Euler’s method is unsuitable for stiff equations and suggest an improvement.
Answer
Stiff systems require extremely small step sizes for stability. Implicit or higher-order methods such as RK4 are more suitable.