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∞

Direction Fields and the Geometric View of ODEs

Why does $y' = y - x^2$ solve in elementary functions while the nearly identical $y' = x - y^2$ has no solution at all?

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Your summary note

1. Standard form $y' = f(x,y)$ of a first-order ODE
  Write down that first order means only the first derivative appears, and that standard form isolates the derivative on the left with everything else on the right.
2. Solvable versus unsolvable simple-looking equations
  Record that $y' = x/y$ separates, $y' = y - x^2$ is linear and solvable, while the nearly identical $y' = x - y^2$ has no elementary solution, motivating a geometric approach.
3. Analytic view paired with the geometric view
  State that the equation $y' = f(x,y)$ corresponds to a direction field and that a solution $y_1(x)$ corresponds to an integral curve.
4. Direction fields from line elements and integral curves
  Write that at each point $(x,y)$ a line element has slope $f(x,y)$ , and that an integral curve is tangent to the line element at every one of its own points.
Attempt 1 of 2