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∞

Integral Curves and the Isocline Method

Why does picking the slope first and drawing the curve $f(x,y) = C$ beat a computer's point-by-point lattice?

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Your summary note

1. Theorem linking solutions and integral curves
  State that $y_1(x)$ solves $y' = f(x,y)$ if and only if its graph is an integral curve, then reproduce the proof translating both sides into $y_1'(x) = f(x, y_1(x))$ .
2. How a computer draws a direction field
  Describe the lattice method of picking equally spaced points and computing $f(x,y)$ at each to draw a line element there, and note it is wasteful for a human.
3. The isocline method for humans
  Record picking a slope $C$ first and plotting the isocline $f(x,y) = C$ , an ordinary level curve rather than the differential equation, then hanging line elements of slope $C$ along it.
4. Drawing convention for isoclines versus solutions
  Note that isoclines are drawn dashed while only integral curves are drawn solid, and give the $C=0$ and $C=-1$ isoclines as examples.
Attempt 1 of 2