Underdetermined ODEs

• Complex ODE System
• Matrix Exponentiation

Finding eigenvectors

\(A = n \times n\) matrix, the task of finding eigenvalues and eigenvectors is a linear algebra problem:

\begin{equation} A v = \lambda v \end{equation}

Finding specific solutions to IVPs with special substitution

For some:

\begin{equation} \begin{cases} x’ = Ax \\ x(t_0) = x_0 \end{cases} \end{equation}

we can leverage the first task:

1. find \(v\), \(\lambda\) for \(A\)
2. guess \(x = u(t)v\), this is “magical substitution”
3. and now, we can see that \(x’ = u’v = A(uv) = \lambda u v\)
4. meaning \(u’ = \lambda u\)
5. finaly, \(u(t) = ce^{\lambda} t\)

Eigenbasis case

Suppose \(A\) has a basis of eigenvectors, and real eigenvalues. We can write its entire solution set in terms of these basis eigenvectors:

Review

For Linear Constant-Coefficient Equation that are homogeneous, we can solve it generally in terms of some matrix \(A\) as:

\begin{equation} x’ = Ax \end{equation}

if \(A\) has enough eigenvectors, we can just write out \(y(t) = c_1 e^{\lambda_{1}t} v_1 + … + c_{n}e^{\lambda_{n}t} v_{2}\)

But, if we don’t, we can use matrix exponentiation

Content

• eigensolutions

We’ve gone over Heat Equation, Wave Equation, and let’s talk about some more stuff.

• damped heat equation
• damped wave equation
• two-dimensional heat equation

What if, Fourier Series, but exponential?

This also motivates Discrete Fourier Transform.

Also Complex Exponential.

Review

Recall again that if we have a periodic function, we’ve got:

\begin{equation} f(x) = \sum_{k=0}^{\infty} a_{k} \sin \qty( \frac{2\pi k}{l} x) + b_{n} \cos \qty( \frac{2\pi k x}{L}) \end{equation}

We note that this breaks individually into the sign and cosine series depending of the function’s oddness.

Complex Fourier Series

This will begin by feeling like a notation rewrite: