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∞

Euler's Method and Its Three Equations

How does following one line element at a time turn a direction field into the three recursive equations a computer actually runs?

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Your summary note

1. The initial value problem (IVP)
  Write the pair $y' = f(x,y)$ and $y(x_0) = y_0$ , name the two parts, and note that numerical methods are how such problems are actually solved in practice.
2. Euler's method geometrically
  Describe starting at $(x_0, y_0)$ , following the line element of slope $f(x,y)$ for one step of size $h$ , recomputing the slope at the new point, and repeating to produce a broken-line approximation.
3. Deriving the three Euler equations
  Complete the triangle at $(x_n, y_n)$ with slope $A_n$ , horizontal side $h$ and vertical side $h A_n$ , then write out $h$ , $y_{n+1} = y_n + h A_n$ , and $A_n = f(x_n, y_n)$ .
Attempt 1 of 2