Differential Equations and Numerical Methods Review

A set of 15 vocabulary flashcards covering the fundamental concepts of differential equations, projectile physics, and numerical approximation methods as discussed in the lecture.

Differential Equation

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

Order of a Differential Equation

The highest order of derivative that appears in the polynomial; for example, a second derivative makes it a second order differential equation.

General Solution

A solution where the constant $C$ has not yet been chosen, representing an infinite set of functions that satisfy the equation.

Particular Solution

A specific solution chosen from the general solution to satisfy a given condition, such as passing through a point $(2, 7)$.

Initial Value Problem

A differential equation accompanied by an extra condition, such as the value at time $t = 0$, which allows for the identification of a unique function.

Anti-derivative

The process of integration used to solve differential equations by reversing the derivative, for example turning $y' = 2 - x$ into $y = 2x - \frac{x^2}{2} + C$.

Gravity

The constant acceleration towards the Earth, quantified on the surface as $-9.8\,m/s^2$.

Velocity

The first derivative of the position function with respect to time, often appearing as $v(t)$ or $s'(t)$. Davis.

Acceleration

The derivative of the velocity function with respect to time ($v'(t)$), which describes the rate of change of velocity.

Initial Velocity

The velocity of an object at the start time, denoted as $v(0)$, used as a condition to find the solution for a velocity function.

Direction Field

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

Equilibrium Solutions

Constant solutions where the derivative equals zero ($y' = 0$), creating horizontal lines in a direction field that are found by setting factors of the differential equation to zero.

Linear Approximation

The use of a tangent line at a known point to estimate the value of a function at a nearby, unknown value.

Linearization

The specific function $L(x) = f'(a)(x - a) + f(a)$ used to calculate approximations near a point $a$.

Euler's Method

A numerical method that uses repeated linearizations and small step sizes to approximate points along the curve of a differential equation's solution.