We have a function:

\begin{equation} |x|+|y|\frac{dy}{dx} = \sin \left(\frac{x}{n}\right) \end{equation}

We are to attempt to express the solution analytically and also approximate them.

To develop a basic approximate solution, we will leverage a recursive simulation approach.

We first set a constant \(N\) which in the \(N\) value which we will eventually vary.

N = 0.5

We can get some values by stepping through \(x\) and \(y\) through which we can then figure \(\frac{dy}{dx}\), namely, how the function evolves.

Generative Classifier

A Generative Classifier builds a good model of a class, and use that to assign how “class-y” is that image.

For instance, to categorize cats vs. dogs, we build a cat model and dog model. To classify, then, we see if a particular image is more “cat-y” or “dog-y”.

Discriminative Classifier

A Discriminative Classifier observes the differences between two classes, instead of trying to model each one.

A Differential Equation is a function-valued algebreic equation whose unknown is an entire function \(y(x)\), where the equation involves a combination of derivatives $y(x), y’(x), …$.

See Differential Equations Index

and Uniqueness and Existance

Differential Equations. math53.stanford.edu.

Logistics

Prof. Rafe Mazzeo

TAs

• Rodrigo Angelo
• Zhenyuan Zhang

Assignments

• Pre-lecture reading + questionnaire
• PSets: Wed 9A
• 2 Midterms + 1 Final: wk 4 + wk 7, Thurs Evening; Tuesday 12:15

Review

1. it suffices to study First Order ODEs because we can convert all higher order functions into a First Order ODEs
2. homogeneous linear systems \(y’=Ay\) can be solved using eigenvalue, matrix exponentiation, etc. (recall that special cases exists where repeated eigenvalues, etc.)
3. inhomogeneous systems \(y’ = Ay +f(t)\) can be solved using intergrating factor or variation of parameters method
4. general analysis of non-linear \(y’=f(y)\): we can talk about stationary solutions (1. linearize each \(y_0\) stationary solutions to figure local behavior 2. away from stationary solutions, use Lyapunov Functions to discuss), or liapenov functions
5. for variable-coefficient ODEs, we decry sadness and Solving ODEs via power series

Content

• Ordinary Differential Equations
• Partial Differential Equations

What we want to understand: