Key Sequence

Notation

New Concepts

• tying into Separated Equations: \(y’ = f(t,y)\) which are the most nicest. Recall that there was two special cases: seperable and autonomous ODEs.
  • if we can write in terms of elementary function, good times
  • if we can’t do it in terms of elementary functions, we can use qualitative analysis t(slope field, etc.)
• recall again Newton’s Law of Cooling
• phase line and stability (ODEs)

Important Results / Claims

• autonomous First Order ODEs’ solutions do not cross; as in, if there are two solutinos \(y_1\) and \(y_2\), their curves never intersect.
• one and exactly one solution exist for every point of an IVP
• autonomous ODEs level off at stationary curves

Questions

Interesting Factoids

Key Sequence

Notation

New Concepts

• complex number

Recall also \(|z| = \sqrt{\bar{z}z}\)

• Euler’s Equation

Important Results / Claims

• complex numbers, fundamentally, are a way of *multiplying in \(\mathbb{R}^{2}\)
• scaling by reals will result in scaling up, and multiplying by complex will result in rotation.

Questions

Interesting Factoids

Key Sequence

Notation

New Concepts

• Second-Order Linear Differential Equations
  • superposition principle and functional independence
• Newton’s First Law of Motion

Important Results / Claims

• finding independent solutions of second-order constant-coefficient linear ODEs
• homogeneous constant-coefficient second order linear ODE
• Uniqueness and Existance of second order
• superposition principle

Questions

Interesting Factoids

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