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∞

Where Existence and Uniqueness Fail

How can every line through $(0,1)$ solve one equation, while points on the $y$-axis have no solution at all?

Loading…

←Previous Existence and Uniqueness: One Curve Through a Point Next Euler's Method and Its Three Equations →

Your summary note

1. Solving $x\,\frac{dy}{dx} = y - 1$ by separation
  Separate to $\frac{dy}{y-1} = \frac{dx}{x}$ , integrate to $\ln|y-1| = \ln|x| + c_1$ , and absorb the absolute values with $C = \pm e^{c_1}$ to reach $y = 1 + Cx$ .
2. The fan of lines through $(0,1)$
  Describe the solutions as all lines with $y$ -intercept $1$ and any slope, and note that off the $y$ -axis exactly one line passes through each point.
3. Where existence and uniqueness fail
  Record that on the $y$ -axis no solution passes (existence fails) while at $(0,1)$ every line passes (uniqueness fails).
4. No violation of the Existence and Uniqueness Theorem
  Rewrite in standard form $\frac{dy}{dx} = \frac{y-1}{x}$ , note $f(x,y)$ is undefined at $x=0$ so the hypotheses fail there, and state this is how failure happens in ordinary practice.
Attempt 1 of 2