Key Sequence

Notation

New Concepts

• First Order ODEs
  • order of equations
  • linear vs. non-linear equations
  • homogeneous vs. inhomogeneous equations
  • linear systems
• Newton’s Law of Cooling

Important Results / Claims

• superposition principle
• Fundamental Theorem of Calculus

Questions

Interesting Factoids

Key Sequence

Notation

New Concepts

• First Order ODEs
  • autonomous ODEs
  • seperable ODEs
• initial value problems
• interval principle

Important Results / Claims

• division method
• general solution to y’(t) = ry(t)
• IMPORTANT: one and exactly one solution exist for every point of an IVP

Questions

Interesting Factoids

Key Sequence

Notation

New Concepts

• level set
• Newton’s Law of Cooling
• logistic equations

Important Results / Claims

• case study: pietri dish

Questions

Interesting Factoids

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