Introduction to Differential Equations and Families of Curves

Vocabulary practice flashcards covering fundamental definitions, classifications, rules for elimination of arbitrary constants, and geometric representations of families of curves in differential equations.

Differential Equation

An equation containing the derivatives or differentials of one or more dependent variables with respect to one or more independent variables.

Independent Variable

The variable in a differential equation with respect to which derivatives are taken.

Dependent Variable

The variable in a differential equation whose derivatives occur in the equation.

Ordinary Differential Equation

An equation which contains only the derivatives of one or more dependent variables with respect to a single independent variable.

Partial Differential Equation

An equation involving the partial derivatives of one or more dependent variables of two or more independent variables.

Order of a Differential Equation

The order of the highest-order derivative term involved in the differential equation.

Degree of a Differential Equation

The algebraic degree of the highest-order derivative in the differential equation.

Linear Ordinary Differential Equation

An ordinary differential equation where the dependent variable $y$ and its derivatives occur to the first degree only, no products of $y$ and/or its derivatives are present, and no transcendental functions of $y$ or its derivatives occur.

Non-Linear Ordinary Differential Equation

An ordinary differential equation that is not linear.

Properties for Elimination of Arbitrary Constants

1. The order of the differential equation must equal the number of arbitrary constants in the equation. 2. Differentiation must be consistent with the given relation. 3. The differential equation must be free from any arbitrary constant.

Family of Lines Passing Through the Origin

A one-parameter family of lines represented equationally by $y = cx$.

Family of Parabolas with Axis along the Y-axis

A one-parameter family of parabolas represented equationally by $y = ax^2$.

Family of Concentric Circles Centered at the Origin

A one-parameter family of concentric circles represented equationally by $x^2 + y^2 = c^2$.

Family of Circles with Centers on the Y-axis

A two-parameter family of circles represented equationally by $x^2 + (y - k)^2 = c^2$.

Family of Circles with Centers on the X-axis

A two-parameter family of circles represented equationally by $(x - h)^2 + y^2 = c^2$.

Family of Circles Centered on $y = x$ Passing Through the Origin

A one-parameter family of circles represented equationally by $(x - a)^2 + (y - a)^2 = 2a^2$.

Family of Lines with Equal Intercepts

A one-parameter family of lines represented equationally by $x + y = c$ or $\frac{x}{a} + \frac{y}{a} = 1$.

Family of Lines with Equal Slope and Y-intercept

A one-parameter family of lines represented equationally by $y = cx + c$.

Family of Parabolas Opening Right with Vertex and Focus on the X-axis

A two-parameter family of parabolas represented equationally by $(y - k)^2 = 4a(x - h)$ where $a > 0$.

Family of Lines Passing Through a Fixed Point $(h, k)$

A one-parameter family of straight lines represented equationally by $y - k = m(x - h)$, where $h$ and $k$ are fixed constants and $m$ is the parameter.

Family of Circles Tangent to the X-axis with Center at $(h, k)$

A two-parameter family of circles represented equationally by $(x - h)^2 + (y - k)^2 = k^2$, where the radius equals $k$.

Family of Hyperbolas Centered at the Origin with Vertices at $V(\pm 1, 0)$

A family of hyperbolas represented equationally by $x^2 - y^2 = c$, where $c > 0$.

Family of Circles of Radius 1 with Center on the Line $y = x$

A one-parameter family of circles represented equationally by $(x - a)^2 + (y - a)^2 = 1$.