2.6 KiB
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:
[ a_n(t)y^{(n)}(t) + a_{n-1}(t)+y^{n-1}(t) + \ldots + a_1(t)y'(t) + a_0(t)y(t) = g(t) ]
Note that a_n(t) does not depeond 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 inverval \alpha < t < \beta are any funcion(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 coniditions 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).