more on Fourier Series.

decomposition of functions to even and odd

Suppose we have any function with period \(L\) over \([-\frac{L}{2}, \frac{L}{2}]\), we can write this as a sum of even and odd functions:

\begin{equation} f(x) = \frac{1}{2} (f(x) - f(-x)) + \frac{1}{2} (f(x) + f(-x)) \end{equation}

And because of this fact, we can actually take each part and break it down individually as a Fourier Series because sin and cos are even and odd parts.

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