Numerical ODEs
Objective. Compare Euler and Runge-Kutta methods.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Numerical methods for ordinary differential equations approximate the solution of y′ = f(t,y) at discrete steps when no closed form is available. Euler's method uses yₙ₊₁ = yₙ + h f(tₙ,yₙ), a first-order approximation with error O(h) per step, while the classical fourth-order Runge-Kutta method averages four slope estimates per step to achieve error O(h⁴), far more accurate for the same step size. For y′ = y, y(0) = 1, h = 0.1, Euler gives y₁ = 1.1, while RK4 gives y₁ ≈ 1.10517, matching e^0.1 to five decimal places. Smaller step size always improves Euler's accuracy but at the cost of more computation.
Worked example
Use Euler's method with step size h = 0.5 to estimate y(0.5) for the ODE y′ = 2y, y(0) = 1.
- 1Write Euler's update: yₙ₊₁ = yₙ + h f(tₙ,yₙ) with f(t,y) = 2y.
- 2Substitute n = 0: y₁ = y₀ + h(2y₀) = 1 + 0.5(2×1).
- 3Compute: y₁ = 1 + 1 = 2.
- 4Check: this is one step from t=0 to t=0.5, so y(0.5) ≈ 2.
Answer: y(0.5) ≈ 2
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Start practisingCommon mistake. Students assume halving the step size h always halves Euler's error; in fact Euler's global error is proportional to h, but using too large an h on a stiff equation can make the numerical solution diverge entirely rather than merely lose accuracy.
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