Differential Equations

Math 4B

First Lecture Activity

Match well known functions with the DEs that describe them

Slope Fields

Matching Curves and Initial Conditions

Just Product Rule

Collapsing y' and y with the product rule

Integrating Factor

Finding the right integrating factor first

Applications

Randomized Practice Problems: Brine Mixing, Exponential Growth and Decay, Logistics

Distinct Real Roots

Constant Coefficients with Distinct Real Roots

Complex Conjugate Roots

Constant Coefficients with Complex Conjugate Roots

Repeated Roots

Constant Coefficients with Repeated Roots

Reduction of Order

Recovering the full solution set given one homogeneous solution

Cauchy-Euler Equations

The Monomial Analogue of the Constant Coefficients Equations

Exponential Functions

Finding the right constant multiple

Polynomial Functions

Using either Iterative Refinement or Undetermined Coefficients

Sine and Cosine Functions

Using either Iterative Refinement or Undetermined Coefficients

Exponential Functions with Conflicting Roots

The desired exonential is alredy a homogeneous solution

Variation of Parameters

A Determinant and Integration Heavy Method for finding Particular Solutions

Extra Variation of Parameters Problems (external reference)

Warning: These problems can be very tedious. Practice these sparingly if at all.

Damped Motion

Over, Under, and Critically Damped Harmonic Motion

Laplace Transform (external reference)

Use the 6.1 problems that practice transforming back and forth

What Are Eigenvectors?

Getting a better understanding

How Do We Use Eigenvectors?

Using eigen-axes to model matrices

Computing Eigen-coordinates

A very short guide

Computing Eigenvectors

How to Compute Eigenvalues and Eigenvectors of a Matrix

DE Connection

Using eigenvectors in differential equations

Review Worksheet

A comprehensive pdf with the definition through diagonalization and applications

Homogeneous System with Distinct Real Roots

More Precisely: Method works for any 2x2 matrix that is diagonalizable

Inner Products and Approximations

Approximating e^x with quadratic functions.

Sine and Cosine Properties

The Special Properties of Sine and Cosine following from the previous section

Computing Fourier Coefficients

Computations and Visual Examples

Computing Sine Coefficients

Find the sine coefficients to best approximate each function