Introduction to Differential Equations and Numerical Methods

Flashcards covering the terminology and core concepts of differential equations, physical applications, and numerical approximation methods like Euler's Method.

Differential Equation

An equation that involves a function $y$ and one or more of its derivatives.

Order of a Differential Equation

The highest order of derivative that appears in the equation, similar to how the degree is the highest power in a polynomial.

First Order Differential Equation

A differential equation where the largest derivative that appears is the first derivative, denoted as $y'$.

General Solution

A solution to a differential equation that represents a family of functions, usually containing an unknown constant $C$.

Particular Solution

A specific solution to a differential equation that is found by choosing a value for the constant $C$, often to satisfy a given point or condition.

Initial Value Problem

A differential equation accompanied by an initial condition, such as $y(x_0) = y_0$, which allows for the determination of a unique solution.

Velocity ($v(t)$) relationship

In physics, the velocity is the derivative of the position function ($s'(t)$) and the anti-derivative of the acceleration function.

Acceleration due to Gravity

The derivative of velocity ($V'(t)$), which at the Earth's surface is approximately $9.8\,m/s^2$.

Direction Field

A graphical representation of all possible solutions to a differential equation using arrows to show the slope of the tangent line at various points.

Equilibrium Solutions

Solutions where the derivative $y'$ is zero, resulting in horizontal tangent lines and a stable value for the function.

Linearization

A method using the tangent line to a function at a specific point to approximate function values for nearby points, defined by $L(x) = f'(a)(x - a) + f(a)$.

Euler's Method

A numerical method used to approximate the solution of an initial value problem by taking small steps along successive tangent lines.

Separable Differential Equation

A type of differential equation where variables can be separated and solved by integrating both sides.

Slope of the Tangent Line

The value described by the derivative $y'$ in a differential equation at a specific point $(x, y)$.

Chain Rule in Derivatives

A rule used when taking derivatives of composite functions, such as the derivative of $3e^{2x}$ becoming $6e^{2x}$.