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∞

First Order Linear: Standard Form and the Sign of p

Why does writing $y' = -py + q$ instead of $y' + py = q$ flip the sign of $p$ and change the solution's behavior entirely?

Loading…

←Previous Singular Points: Where Every Method Fails Next The Conduction-Diffusion Model: Newton's Law of Cooling →

Your summary note

1. Linearity in $y$ and $y'$ , and homogeneity
  Write $a(x)y' + b(x)y = c(x)$ , set it beside the algebraic $a y_1 + b y_2 = c$ with $y'$ treated as a second variable, and record when the equation is homogeneous.
2. Standard linear form $y' + p(x)y = q(x)$
  Divide $a(x)y' + b(x)y = c(x)$ through by $a(x)$ , rename the coefficients $p = b/a$ and $q = c/a$ , and write out the resulting standard form.
3. The sign trap in $y' = -p(x)y + q(x)$
  Record both rival forms side by side, note that $y' + p(x)y = q(x)$ is the one used throughout, and state that a flipped sign of $p$ produces solutions with totally different behavior.
4. Always solvable, and where linear equations arise
  State that a first-order linear equation can always be solved, and list conduction-diffusion (temperature-concentration), mixing, radioactive decay, bank interest and motion problems as the models it governs.
Attempt 1 of 2