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∞

Convexity and the Sign of Euler's Error

How can differentiating $y' = f(x,y)$ tell you whether Euler's answer is too high or too low, without ever solving it?

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Your summary note

1. Euler too high or too low from curvature
  Record that a convex solution curve makes Euler too low and a concave curve makes Euler too high, with the geometric picture of the broken line cutting the corner across successive parallel line elements.
2. Convexity as the sign of $y''$
  State that convex means $y'' > 0$ and concave means $y'' < 0$ , reducing the question to the sign of the second derivative at the starting point.
3. Getting $y'' < 0$ from the equation by differentiating
  Show how to differentiate $y' = f(x,y)$ using the chain rule and evaluate at the initial point, working through $y' = x^2 - y^2$ to get $y''(0) = 2$ .
4. Reliability only nearby
  Note the caveat that the curve may later switch from convex to concave, so the too-low or too-high conclusion holds only while the approximation stays near the starting point.
Attempt 1 of 2