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∞

Isocline Corridors and a Trap for Solutions

How can two parallel isoclines form a corridor that an integral curve can enter but never escape?

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Your summary note

1. Parallel isoclines of $y' = 1 + x - y$
  Set $1 + x - y = C$ and solve to get $y = x + 1 - C$ , then note these are parallel lines of slope 1 differing only in intercept, and plot a few with their line elements.
2. The line $y = x$ as isocline and integral curve
  Show that on the $C = 1$ isocline $y = x$ the elements have slope 1 matching the line, and verify $y' = 1$ equals $1 + x - x = 1$ so the line is both isocline and solution.
3. The corridor lobster trap
  Take the corridor between the $C = 0$ line $y = x+1$ and the $C = 2$ line $y = x-1$ , and argue that a curve cannot cross out the top or the bottom, so once inside it stays.
4. Geometry as the only available answer
  Record that this equation cannot be separated, so no analytic solution exists, and the direction-field geometry is the whole answer and comes quickly.
Attempt 1 of 2