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∞

Existence and Uniqueness: One Curve Through a Point

Why can two integral curves never even touch, when every point of the plane carries just one line element?

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Your summary note

1. Principle 1: integral curves cannot cross at an angle
  State the Intersection Principle part (i) and argue from the direction field giving a single line element with one value $f(x_0,y_0)$ at each point.
2. Asymptotic corridor and the exact solution $y=x$
  Describe how trapped curves close in on $y=x$ for $y'=1+x-y$ , and verify $y=x$ solves it by substituting to get $1=1$ .
3. Principle 2: integral curves cannot be tangent
  Record that touching is also forbidden, so trapped curves only become asymptotic to $y=x$ rather than joining and riding along it.
4. Existence and Uniqueness Theorem and its hypotheses
  Write that $y'=f(x,y)$ has one and only one solution through $(x_0,y_0)$ , distinguishing 'one' as existence from 'only one' as uniqueness, and list continuity of $f$ and of $\partial f/\partial y$ .
Attempt 1 of 2