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∞

Constant Coefficients: Steady State and Transient

Why does the initial temperature $T_0$ stop mattering once $e^{-kt}$ decays, leaving only the steady-state integral?

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Your summary note

1. Standard form $\frac{dT}{dt} + kT = kT_e$
  Write the conduction equation in standard form with $kT_e$ on the right rather than $T_e$ , and carry out the units comparison of the two sides.
2. The integrating factor $e^{kt}$
  Integrate $k\,dt$ , multiply through, write the compact form $(e^{kt}T)' = kT_e(t)e^{kt}$ , and integrate to $T = e^{-kt}\int kT_e(t)e^{kt}\,dt + Ce^{-kt}$ .
3. Definite-integral solution with $T(0) = T_0$
  Redo the integral from $0$ to $t$ in the dummy variable $t_1$ , then set $t = 0$ and record the resulting value $C = T_0$ .
4. Steady-state term and transient $T_0 e^{-kt}$
  Name the two terms of $T = e^{-kt}\int_0^t kT_e(t_1)e^{kt_1}\,dt_1 + T_0 e^{-kt}$ and write the $t \to \infty$ behaviour of each for $k > 0$ .
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