Section 1 - Basic Concepts
Section 1.1 - Definitions
This section is from Paul's Online Math Notes.
Definition. A differential equation is an equation that describes a function in terms of its derivatives. Examples of differential equations include Newton's Laws, among others.
Definition. The order of a differential equation is the largest derivative present in the equation with a non-zero constant.
Definition. A differential equation that only involves derivatives with respect to one variable is called an ordinary differential equation (ODE).
Definition. A differential equation that describes a function in terms of derivatives with respect to more than one linearly-independent variable is called a partial equation.
Definition. A linear differential equation is any differential equation that cn be written in the following form:
Note that \(a_n(t)\) does not depend on any derivative of \(y\), so the presence of terms such as \(e^y\) or \(\sqrt{y'}\) signal that the equation is nonlinear.
Definition. The solution(s) to a differential equation over an interval \(\alpha < t < \beta\) are any function(s) \(y(t)\) that satisfy the differential equation.
Definition. The initial conditions are a condition or set of conditions that constrain the possible solution sets.
Definition. An Initial Value Problem is a differential equation along with the appropriate boundary or initial conditions.
Definition. The integral of validity for a solution to a differential equation is the largest possible interval containing the initial conditions for which the solution is valid.
Definition. The general solution to a differential equation is the most general form a solution to a differential equation can take without requiring the initial conditions.
Definition. The actual solution to a differential equation is the specific solution that satisfies the differential equation and the boundary conditions.
Definition. A solution is said to be explicit if it can be written in the form \(y = y(t)\). Otherwise, it is said to be implicit.
Section 1.2 - Directional Fields
This section is from Paul's Online Math Notes.
Definition. A directional field is the graph of a \(t\) vs. \(y(t)\), with vectors drawn at each point with a slope corresponding to \(y'(t)\). Notably, each arrow will be pointed right (towards increasing \(t\)).