Differential Equations Review

A collection of 50 vocabulary flashcards covering differential equations concepts, types, and specific solutions based on the lecture notes.

Blasius equation

A third order nonlinear ordinary differential equation defined by the expression $2 \frac{d^3 f}{d \theta^3} + f \frac{d^2 f}{d \theta^2} = 0$.

Order of a Partial Differential Equation

The highest derivative present in the equation; for example, the equation $\frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + \frac{\partial \phi}{\partial x} + \frac{\partial \phi}{\partial y} = 0$ is of order 2.

Degree of a Partial Differential Equation

The power or exponent to which the highest-order derivative is raised; the equation $\frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + \frac{\partial \phi}{\partial x} + \frac{\partial \phi}{\partial y} = 0$ has degree 1.

Bernoulli Equation

A differential equation of the form $y' + P(x)y = R(x)y^{\alpha}$.

Linear Differential Equation Case of Bernoulli

The classification of the Bernoulli equation $y' + P(x)y = R(x)y^{\alpha}$ when the exponent $\alpha = 1$.

Euler’s Equation

A second-order differential equation of the form $x^2 y'' + Axy' + By = 0$ where $A$ and $B$ are constants.

Convolution Theorem

A theorem in Laplace transforms stated as $L[f * g](s) = F(s)G(s)$.

General Solution of $xdy = ydx$

A family of lines passing through the origin.

Integrating Factor for $(2y - x^3)dx + xdy = 0$

A function, which for this specific equation is $x$, used to facilitate solving the linear differential equation.

First Shifting Formula

A Laplace transform property that relates requested functions to their exponentially shifted counterparts.

Second Shifting Formula

An advanced property of Laplace transforms involving the shifting of functions in the time domain.

Separable Differential Equation

A differential equation where variables can be isolated on opposite sides, such as $u' = e^{-v}$ or $du = e^{-v} dv$.

Inverse Transform

The process symbolized by $L^{-1}[F(s)]$ to return a transformed function to its original domain.

Initial Condition

A specified value of the solution at a specific point, such as $y(1) = 6/5$ or $y(0) = 1$, used to find a particular solution.

General Solution of $dy + 27y = 0$

The exponential function $y = Ce^{-27t}$.

Solution of $y'' + 5y' + 6y = 0$ with $y(0) = 0$ and $y'(0) = 1$

The particular solution given by $y = e^{-2x} - e^{-3x}$.

Orthogonal Trajectories of $\mathbf{y = Ce^{-x}}$

A family of curves that intersect the given family at right angles, specifically $y^2 = 2x + C$.

Differential Equation of lines through the origin

The first-order differential equation represented by $ydx - xdy = 0$.

Orthogonal Trajectories of $y^2 = 4cx$

The family of curves represented by the equation $y^2 + 2x^2 = C$.

Solution of $y' + 2y = 1$ with $y(0) = 1$

The function $y = \frac{1}{2} + \frac{1}{2} e^{-2t}$.

Wave Equation

A category of partial differential equations, often distinguished from Euler or Bernoulli equations.

Ricatti Equation

A nonlinear first-order ordinary differential equation often listed alongside Wave and Bernoulli equations.

Characteristic Solution for $p = 4$ and $m = 3$

In second-order differential equations, a solution form like $y = (C_1 + C_2 x) e^{-2x}$.

Solution of $e^{-y} dy = x^5 dx$

The resulting relation $e^y = \frac{x^6}{6} + C$ after integration.

Family of Circles

A geometric family represented by equations like $x^2 + y^2 = r^2$, contrasted with lines through the origin.

Family of Parabolas

A geometric family of curves such as $y^2 = 4cx$, often studied for its orthogonal trajectories.

Linearity

The property describing a differential equation where the dependent variable and its derivatives appear only in the first power.

Elliptic Equation

A classification of second-order partial differential equations based on the discriminant of its coefficients.

Hyperbolic Equation

A classification of second-order partial differential equations often associated with wave propagation.

Solution for $x \frac{dy}{dx} + y = 4x$ at $y(1) = 6/5$

The function $y = \frac{5x^2 + 1}{5x}$.

General Solution of $\frac{dy}{dx} + \frac{2}{x}y = \cos(x)$

The function $y = \frac{x \sin(x) + \cos(x) + C}{x^2}$.

Integrating Factor for $x^3 dx + 3xy^2 dy = 0$

A mathematical term, such as $x^{-2}$, applied to verify or create an exact differential equation.

Solution for $y' = x - xy$

The function $y = 1 + Ce^{-x^2/2}$.

Exact Differential Equation

An equation of the form $M(x,y)dx + N(x,y)dy = 0$ where the partial derivative of $M$ with respect to $y$ equals the partial derivative of $N$ with respect to $x$.

Substitution for Bernoulli Equations

The technique of using $u = y^{1-\alpha}$ to transform a nonlinear equation into a linear one.

Particular Solution

A solution to a differential equation that is free of arbitrary constants, obtained by applying initial or boundary conditions.

Ordinary Differential Equation (ODE)

A differential equation involving functions of only one independent variable and their derivatives.

General Solution of $(y - 2x)dx + (y + 2x)dy = 0$

The expression $x^2 - xy - y^2 = C$ or similar implicit forms derived from homogeneous equations.

Family of Parallel Lines

A set of lines with the same slope but different intercepts, such as $y = mx + C$.

Boundary Value Problem

A problem where the dependent variable or its derivatives are specified at more than one point.

Coefficient of $y'$ in Euler's Equation

The term $Ax$ where $A$ is a constant multiplier of the first derivative.

Characteristic roots for $y'' + 5y' + 6y = 0$

The values $-2$ and $-3$ used to form the basis of the general solution.

Solution of $\sin(x) y' + \cos(x) y = \ln(x)$

The result $y = \csc(x) (x \ln(x) - x + C)$.

Derivative Order Notation

The use of primes (e.g., $y''$) or Leibniz notation (e.g., $\frac{d^2 y}{dx^2}$) to denote the level of differentiation.

Calculus of Exponential Forms

The integration rule where $\int e^{kt} dt = \frac{1}{k}e^{kt} + C$.

Implicit Solution

A solution to a differential equation where the dependent variable is not isolated on one side of the equation.

Dependent Variable

The variable being solved for in a differential equation, typically $y$ or $f(x)$.

Independent Variable

The variable with respect to which derivatives are taken, typically $x$, $t$, or $\eta$.

Constant of Integration

The arbitrary constant, typically denoted as $C$, added to the general solution of an indefinite integral.

Family of Curves

A set of curves where each curve is defined by assigning a specific value to a constant in a general equation.