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∞

The Conduction-Diffusion Model: Newton's Law of Cooling

Why is Newton's law of cooling written $k(T_e - T)$ rather than $k(T - T_e)$, and why is salt diffusion the very same equation?

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Your summary note

1. Newton's law of cooling, $\frac{dT}{dt}=k(T_e-T)$
  Set up the chamber in the bath with internal temperature $T$ and external $T_e$ , write the equation, reproduce the sign argument that fixes the order $T_e-T$ and keeps the conductivity $k$ positive, and add $T(0)=T_0$ .
2. Diffusion across a semipermeable membrane, $\frac{dC}{dt}=k_1(C_e-C)$
  Write the same model with inside concentration $C$ , outside concentration $C_e$ and $k_1>0$ , and match each symbol to its counterpart in the conduction equation.
3. Standard linear form, with $k(t)$ and $T_e(t)$ varying
  Rearrange the conduction equation into $\frac{dT}{dt}+kT=kT_e$ , compare it with $y'+py=q$ , then write $\frac{dT}{dt}+k(t)T=k(t)T_e(t)$ and state that this is the general first-order linear equation.
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