Differential Equations

The Geometric View of ODEs

01Direction Fields and the Geometric View of ODEs 02Integral Curves and the Isocline Method 03Perpendicular Isoclines: The Circles of y' = -x/y 04Isocline Corridors and a Trap for Solutions 05Existence and Uniqueness: One Curve Through a Point 06Where Existence and Uniqueness Fail Problem set0/10 Problem set 20/10 MIT problem set0/3 Practice∞

Numerical Methods

01Euler's Method and Its Three Equations 02Euler's Method by Hand: Building the Table 03Convexity and the Sign of Euler's Error 04First Order Error: Halve the Step, Halve the Error 05Improved Euler (RK2): Averaging Two Slopes 06Runge-Kutta 4 and the Price of Accuracy 07Singular Points: Where Every Method Fails Problem set0/10 Problem set 20/10 MIT problem set0/1 Practice∞

First-Order Linear Equations

01First Order Linear: Standard Form and the Sign of p 02The Conduction-Diffusion Model: Newton's Law of Cooling 03The Integrating Factor and the Four-Step Method 04Integrating Factors: Two Mistakes and a Trick Question 05Constant Coefficients: Steady State and Transient Problem set0/10 Problem set 20/10 MIT problem set0/1 Practice∞

First Order Error: Halve the Step, Halve the Error

Why does halving Euler's step size only halve the error, and what does that send you looking for instead?

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←Previous Convexity and the Sign of Euler's Error Next Improved Euler (RK2): Averaging Two Slopes →

Your summary note

1. Definition of error and systematic error
  Write $e =$ true value $-$ Euler value, note the sign convention, and record that convexity or concavity makes the sign predictable, marking it a systematic error.
2. First order error $e \approx Ch$
  State that the error is asymptotically proportional to $h$ to the first power, that this names Euler a first-order method, and record the halve-the-step-halve-the-error rule.
3. First order versus order of the ODE
  Record the distinction that 'first order' here refers to the power of $h$ , not to the order of the differential equation $y' = f(x,y)$ .
4. Toward a better slope
  Note that better methods keep Euler's structure but replace the guaranteed-wrong slope $A_n$ with a corrected slope aimed to land back on the solution curve.
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