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Definition: A differential equation is an equation that relates an unknown function y(t) to its derivatives. It can be expressed in the general form F(t, y(t), y'(t), y"(t), ...) = 0.
Variables:
- Dependent Variable: The unknown function, typically denoted by letters like y.
- Independent Variable: Generally denoted by t (or x), which does not get altered by the function that depends on it.
- It is important to identify these variables correctly based on context.

Understanding Solutions: For example, consider the differential equation y''(t) - y(t) = 0 (where O(t) = 0).
Question Example: Is y(t) = e^t a solution of the ordinary differential equation (ODE)? Testing leads to:
- y' = e^t, and substituting yields:
  - e^t - e^t = 0, which is valid.
- Thus, y(t) = e^t is indeed a solution.
Related Solutions: Explore other forms such as et, e^{-t}, and combinations of these for potential solutions.

Mathematical Significance: They are crucial for understanding various mathematical frameworks, inclusive of algebra and differential geometry.
Modeling and Applications:
- Modeling Definition: A process of describing physical phenomena through mathematical idealizations (often equations).
- Example Application: Consider N(t), representing the number of bacteria on a nutrient plate, starting at N(0) = 1000. The growth rate is modeled to be proportional to current bacteria count: dN/dt = kN.

Equation: Verify the solution N(t) = 1000 e^(kt) using initial conditions and properties:
- Given the parameters, confirm that dN/dt aligns: dN/dt = k(1000 e^(kt)).
- Interpretation: This showcases real-life growth scenarios, prompting the exploration of experimental setups to estimate k and validate modeling accuracy.

Concept of Equilibrium: For an ODE of the form y"(t) = f(t, y(t)), an equilibrium solution refers to a constant solution independent of time (y(t) = k).
- Condition for Equilibrium: For k to be an equilibrium, must satisfy f(t, k) = 0 for all t.
Example: For the equation dy/dt = r(1 - y/k), find the equilibrium solutions:
- Set dy/dt to 0.
- Results show that y = 0 is one solution.

Directional Derivatives: Information on slope behavior can be gleaned without solving directly:
- For fixed points (to, y0) in dy/dt = f(t, y), the slope can indicate whether the function increases, decreases, or remains constant at that juncture.
Example: Consider the differential equation dy/dt = (1 - y). Analyzing this yields insights into the function's behavior across various intervals.