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∞

Euler's Method by Hand: Building the Table

How can two hand-computed rows of a table estimate $y(0.2)$ for an equation that has no elementary solution?

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Your summary note

1. The initial value problem $y' = x^2 - y^2$ , $y(0)=1$ , $h=0.1$
  Record the IVP, the chosen step size, and the fact that this equation has no solution in elementary functions, making it a candidate for a numerical method.
2. The tabular format with columns $n$ , $x_n$ , $y_n$ , $A_n$ , $h A_n$
  Write out the column headings, note that $A_n = x_n^2 - y_n^2$ is the slope at the current point and $h A_n$ is the increment added to $y_n$ , and record the demand to work systematically so mistakes stay findable.
3. Two steps carried through to $y(0.2) \approx 0.82$
  Fill in the rows for $n = 0, 1, 2$ with your own arithmetic — slope $A_n$ , increment $hA_n$ , new $y_{n+1}$ — carrying the table through to $y(0.2) \approx 0.82$ .
Attempt 1 of 2