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∞

Singular Points: Where Every Method Fails

Why can no numerical method reach $y(2)$ for $y' = y^2$, when nothing in the equation says where the solution blows up?

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Your summary note

1. Solving $y' = y^2$ by separation
  Separate variables and carry the integration through to $y = \frac{1}{c - x}$ , then apply $y(0) = 1$ to fix $c = 1$ and locate the blow-up at $x = 1$ .
2. RK4 failing to find $y(2)$
  Describe how the curve from $(0,1)$ rises, blows up at $x=1$ , and returns on a far branch, and record that RK4 with tiny steps marches into the singularity and never reappears.
3. Each solution's private singular point at $x = c$
  State that every solution has its own singularity at $x = c$ whose location the differential equation does not reveal, so one cannot predict in advance where the computation goes bad.
Attempt 1 of 2