Glossary

Boundary Condition

Criticality: 2

A specific point or value that a solution curve of a differential equation must pass through, used to find a particular solution from a general one.

Example:

If a problem states that a solution curve for dy/dx = x passes through (1, 5), then (1, 5) is the boundary condition.

Boundary Conditions

Criticality: 1

Additional information, similar to initial conditions, that specifies the value of a function or its derivatives at the boundaries of a domain, often used for higher-order differential equations.

Example:

In a problem involving heat distribution along a metal rod, specifying the temperature at both ends of the rod would be considered boundary conditions.

Constant of Integration

Criticality: 2

The arbitrary constant, typically denoted by 'C', that arises when performing indefinite integration, representing the family of antiderivatives.

Example:

When you integrate ∫cos(x) dx, the result is sin(x) + C, where C is the constant of integration that accounts for any vertical shift of the antiderivative.

Constant of Proportionality (k)

Criticality: 3

The constant value in the differential equation $\frac{dy}{dt} = ky$ that determines the rate and direction of change (growth if positive, decay if negative).

Example:

If a radioactive isotope has a half-life that results in a decay equation like $\frac{dA}{dt} = -0.01A$ , then -0.01 is the constant of proportionality, indicating its decay rate.

Critical Points

Criticality: 2

Points on a curve where the derivative (gradient) is zero or undefined, often corresponding to local maxima, minima, or saddle points.

Example:

To find the highest or lowest points on a graph, you would typically look for critical points where the slope is zero.

Differential Equation

Criticality: 3

An equation that relates a function with its derivatives, used to model phenomena involving rates of change.

Example:

The equation dy/dt = ky, which describes population growth, is a classic differential equation.

Differential Equation

Criticality: 3

An equation that involves derivatives of a function, expressing a relationship between a quantity and its rates of change.

Example:

The equation dy/dx = ky describes exponential growth, where the rate of change of a population y is proportional to its current size, allowing us to model how a bacterial colony grows.

Differential Equation

Criticality: 3

An equation that relates a function with its derivatives. It describes how a quantity changes with respect to one or more variables.

Example:

The equation $\frac{dy}{dx} = 5y$ is a Differential Equation that models exponential growth.

Differential Equation

Criticality: 3

A mathematical equation that relates a function with its derivatives, describing how a quantity changes with respect to one or more independent variables.

Example:

The equation dP/dt = kP, which models population growth, is a classic differential equation where the rate of change of population (P) is proportional to the population itself.

Differential Equation

Criticality: 2

An equation that relates a function with its derivatives, used to describe how quantities change over time or space.

Example:

To model the velocity of a falling object considering air resistance, one would set up a differential equation that includes terms for gravity and drag.

Exponential Growth and Decay Model

Criticality: 3

A specific type of differential equation, $\frac{dy}{dt} = ky$, used to describe phenomena where a quantity's rate of change is proportional to its current value.

Example:

The exponential growth and decay model can describe the population dynamics of a species, showing rapid increase under ideal conditions or decline due to environmental pressures.

Exponential Models

Criticality: 2

Mathematical models where the rate of change of a quantity is directly proportional to the quantity's current size.

Example:

When studying the spread of a highly contagious disease, epidemiologists often use exponential models to predict how quickly the number of infected individuals will grow.

First Order Differential Equation

Criticality: 3

A differential equation that contains only the first derivative of the unknown function and no higher-order derivatives.

Example:

dy/dt = -0.5y models radioactive decay, where the decay rate depends only on the current amount of the substance, making it a first order differential equation.

First-order differential equation

Criticality: 2

A differential equation that involves only the first derivative of the unknown function.

Example:

$\frac{dy}{dx} = x^2 + y$ is a First-order differential equation because it only contains $\frac{dy}{dx}$ and no higher derivatives.

General Solution

Criticality: 3

The set of all possible functions that satisfy a given differential equation, typically containing one or more arbitrary constants.

Example:

For dy/dx = 2x, the general solution is y = x^2 + C, where C can be any real number, representing an infinite family of parabolas.

General Solution

Criticality: 2

The solution to a differential equation that includes an arbitrary constant (or constants), representing a family of functions that satisfy the equation.

Example:

For $\frac{dy}{dx} = y$ , the General Solution is $y = Ce^x$ , where C can be any real number.

General Solution

Criticality: 3

A solution to a differential equation that contains an arbitrary constant (or constants), representing an infinite family of possible solutions.

Example:

The general solution to dy/dx = 2x is y = x² + C, where 'C' can be any real number, creating a family of parabolas.

Gradient

Criticality: 3

The slope of a curve at a given point, which for a slope field corresponds to the value of dy/dx at that point.

Example:

For a function describing temperature change over time, a positive gradient means the temperature is increasing.

Horizontal Tangent Lines

Criticality: 2

Tangent lines on a slope field that have a slope of zero, indicating points where the derivative dy/dx is equal to zero.

Example:

On a slope field, if you see a row of horizontal tangent lines along y=3, it suggests that y=3 might be an equilibrium solution or a line where local extrema occur.

Initial Condition

Criticality: 3

A specific value of the unknown function (or its derivative) at a particular point, used to determine the unique particular solution from a general solution.

Example:

To find the exact trajectory of a projectile, you need its starting height and initial velocity, which serve as initial conditions for the differential equations describing its motion.

Initial Condition

Criticality: 3

A specific value of the dependent variable at a given starting point (often $t=0$), used to determine the unique constant of integration in a differential equation's solution.

Example:

To predict the future temperature of a cooling object, you need its initial condition, such as its temperature at the moment it was removed from the oven.

Initial Conditions

Criticality: 3

Additional pieces of information provided to determine a specific solution from a family of solutions of a differential equation.

Example:

To find the exact path of a projectile, you need its starting position and velocity, which act as initial conditions for the differential equations describing its motion.

Integration Constant

Criticality: 3

An arbitrary constant, typically denoted by 'C', that is added to the result of an indefinite integral because the derivative of a constant is zero.

Example:

When integrating $\int x dx$ , the result is $\frac{x^2}{2} + C$ , where C is the Integration Constant representing any possible constant value.

Local Minimum/Maximum

Criticality: 2

Points on a function's graph where the function's value is either the smallest (minimum) or largest (maximum) within a specific neighborhood.

Example:

In a roller coaster's path, the lowest dip is a local minimum, and the highest peak is a local maximum.

Particular Solution

Criticality: 3

A specific function that satisfies a differential equation and also meets a given set of conditions, usually an initial or boundary condition.

Example:

If the general solution to dy/dx = 2x is y = x^2 + C, and we know y(0) = 5, then the particular solution is y = x^2 + 5.

Particular Solution

Criticality: 2

A specific solution to a differential equation obtained by using initial conditions to determine the value of the arbitrary constant(s) from the general solution.

Example:

If the General Solution is $y = Ce^x$ and $y(0)=2$ , then $2 = Ce^0 \Rightarrow C=2$ , leading to the Particular Solution $y = 2e^x$ .

Particular Solution

Criticality: 3

A unique solution derived from the general solution of a differential equation by applying specific initial or boundary conditions to determine the value of the arbitrary constant(s).

Example:

If the general solution is y = x² + C and the curve passes through (1, 3), then C = 2, making y = x² + 2 the particular solution.

Rate of change of a quantity is proportional to the size of the quantity

Criticality: 3

A fundamental principle stating that how quickly a quantity increases or decreases is directly dependent on its current amount.

Example:

In a simple interest investment, the interest earned is constant, but in a compound interest account, the more money you have, the more interest you earn, demonstrating how the rate of change of a quantity is proportional to the size of the quantity.

Rates of change

Criticality: 2

How one quantity changes in relation to another, often expressed as a derivative. Differential equations are fundamentally used to model these relationships.

Example:

In physics, the rate of change of an object's position with respect to time is its velocity, and the rate of change of velocity is acceleration.

Regular Grid

Criticality: 1

A set of points arranged in a uniformly spaced pattern, typically used in slope fields to systematically draw tangent lines across the coordinate plane.

Example:

When constructing a slope field by hand, you often choose points like (0,0), (1,0), (0,1), etc., forming a regular grid to calculate slopes.

Separable Differential Equation

Criticality: 3

A type of first-order differential equation that can be written in the form $\frac{dy}{dx} = g(x)h(y)$, allowing the variables to be isolated on opposite sides of the equation.

Example:

The equation $\frac{dy}{dx} = \sin(x) \cdot e^y$ is a Separable Differential Equation because you can write it as $\frac{1}{e^y}dy = \sin(x)dx$ .

Separation of Variables

Criticality: 3

A method used to solve certain first-order differential equations by rearranging terms so that all functions of one variable are on one side of the equation and all functions of the other variable are on the other side.

Example:

To solve $\frac{dy}{dx} = xy$ , you'd use Separation of Variables to get $\frac{1}{y}dy = xdx$ before integrating.

Separation of Variables

Criticality: 2

A technique used to solve certain types of first-order differential equations by rearranging the equation so that all terms involving one variable are on one side and all terms involving the other variable are on the opposite side.

Example:

To solve dy/dx = y/x, you can use separation of variables to get dy/y = dx/x, which can then be integrated on both sides.

Separation of Variables

Criticality: 2

A technique for solving certain types of differential equations by algebraically rearranging terms so that each variable and its differential are on opposite sides of the equation, allowing for independent integration.

Example:

When solving $\frac{dy}{dx} = \frac{x^2}{y}$ , you would use separation of variables to rewrite it as $y , dy = x^2 , dx$ before integrating both sides.

Slope Field

Criticality: 3

A graphical representation of a differential equation, consisting of short tangent lines drawn at various points to illustrate the direction of solution curves.

Example:

When visualizing the flow of water in a river, a slope field could show the direction of the current at different locations.

Solution (to exponential growth & decay model)

Criticality: 3

The explicit function, $y = y_0 e^{kt}$, that satisfies the exponential growth and decay differential equation and its initial condition.

Example:

After solving the differential equation for a growing bacterial colony, the solution $P = P_0 e^{kt}$ allows you to predict the population size at any future time.

Tangent Line

Criticality: 3

A straight line that touches a curve at a single point, indicating the instantaneous direction or slope of the curve at that specific point.

Example:

If you trace the path of a thrown ball, a tangent line at any point on its trajectory would show the direction the ball is moving at that exact moment.

Verifying Solutions

Criticality: 2

The process of substituting a proposed solution and its derivatives back into the original differential equation to confirm that it satisfies the equation.

Example:

To ensure y = e^(2x) is a solution to dy/dx = 2y, you would differentiate y to get dy/dx = 2e^(2x), then substitute e^(2x) for y in the original equation, thereby verifying the solution.

g(x)

Criticality: 2

In a separable differential equation of the form $\frac{dy}{dx} = g(x)h(y)$, $g(x)$ represents the function solely dependent on the variable $x$.

Example:

In the equation $\frac{dy}{dx} = x^2 \cos(y)$ , the g(x) term is $x^2$ .

h(y)

Criticality: 2

In a separable differential equation of the form $\frac{dy}{dx} = g(x)h(y)$, $h(y)$ represents the function solely dependent on the variable $y$.

Example:

In the equation $\frac{dy}{dx} = x^2 \cos(y)$ , the h(y) term is $\cos(y)$ .

Glossary

Boundary Condition

Criticality: 2

A specific point or value that a solution curve of a differential equation must pass through, used to find a particular solution from a general one.

Example:

If a problem states that a solution curve for dy/dx = x passes through (1, 5), then (1, 5) is the boundary condition.

Boundary Conditions

Criticality: 1

Additional information, similar to initial conditions, that specifies the value of a function or its derivatives at the boundaries of a domain, often used for higher-order differential equations.

Example:

In a problem involving heat distribution along a metal rod, specifying the temperature at both ends of the rod would be considered boundary conditions.

Constant of Integration

Criticality: 2

The arbitrary constant, typically denoted by 'C', that arises when performing indefinite integration, representing the family of antiderivatives.

Example:

When you integrate ∫cos(x) dx, the result is sin(x) + C, where C is the constant of integration that accounts for any vertical shift of the antiderivative.

Constant of Proportionality (k)

Criticality: 3

The constant value in the differential equation $\frac{dy}{dt} = ky$ that determines the rate and direction of change (growth if positive, decay if negative).

Example:

If a radioactive isotope has a half-life that results in a decay equation like $\frac{dA}{dt} = -0.01A$ , then -0.01 is the constant of proportionality, indicating its decay rate.

Critical Points

Criticality: 2

Points on a curve where the derivative (gradient) is zero or undefined, often corresponding to local maxima, minima, or saddle points.

Example:

To find the highest or lowest points on a graph, you would typically look for critical points where the slope is zero.

Differential Equation

Criticality: 3

An equation that relates a function with its derivatives, used to model phenomena involving rates of change.

Example:

The equation dy/dt = ky, which describes population growth, is a classic differential equation.

Differential Equation

Criticality: 3

An equation that involves derivatives of a function, expressing a relationship between a quantity and its rates of change.

Example:

The equation dy/dx = ky describes exponential growth, where the rate of change of a population y is proportional to its current size, allowing us to model how a bacterial colony grows.

Differential Equation

Criticality: 3

An equation that relates a function with its derivatives. It describes how a quantity changes with respect to one or more variables.

Example:

The equation $\frac{dy}{dx} = 5y$ is a Differential Equation that models exponential growth.

Differential Equation

Criticality: 3

A mathematical equation that relates a function with its derivatives, describing how a quantity changes with respect to one or more independent variables.

Example:

The equation dP/dt = kP, which models population growth, is a classic differential equation where the rate of change of population (P) is proportional to the population itself.

Differential Equation

Criticality: 2

An equation that relates a function with its derivatives, used to describe how quantities change over time or space.

Example:

To model the velocity of a falling object considering air resistance, one would set up a differential equation that includes terms for gravity and drag.

Exponential Growth and Decay Model

Criticality: 3

A specific type of differential equation, $\frac{dy}{dt} = ky$, used to describe phenomena where a quantity's rate of change is proportional to its current value.

Example:

The exponential growth and decay model can describe the population dynamics of a species, showing rapid increase under ideal conditions or decline due to environmental pressures.

Exponential Models

Criticality: 2

Mathematical models where the rate of change of a quantity is directly proportional to the quantity's current size.

Example:

When studying the spread of a highly contagious disease, epidemiologists often use exponential models to predict how quickly the number of infected individuals will grow.

First Order Differential Equation

Criticality: 3

A differential equation that contains only the first derivative of the unknown function and no higher-order derivatives.

Example:

dy/dt = -0.5y models radioactive decay, where the decay rate depends only on the current amount of the substance, making it a first order differential equation.

First-order differential equation

Criticality: 2

A differential equation that involves only the first derivative of the unknown function.

Example:

$\frac{dy}{dx} = x^2 + y$ is a First-order differential equation because it only contains $\frac{dy}{dx}$ and no higher derivatives.

General Solution

Criticality: 3

The set of all possible functions that satisfy a given differential equation, typically containing one or more arbitrary constants.

Example:

For dy/dx = 2x, the general solution is y = x^2 + C, where C can be any real number, representing an infinite family of parabolas.

General Solution

Criticality: 2

The solution to a differential equation that includes an arbitrary constant (or constants), representing a family of functions that satisfy the equation.

Example:

For $\frac{dy}{dx} = y$ , the General Solution is $y = Ce^x$ , where C can be any real number.

General Solution

Criticality: 3

A solution to a differential equation that contains an arbitrary constant (or constants), representing an infinite family of possible solutions.

Example:

The general solution to dy/dx = 2x is y = x² + C, where 'C' can be any real number, creating a family of parabolas.

Gradient

Criticality: 3

The slope of a curve at a given point, which for a slope field corresponds to the value of dy/dx at that point.

Example:

For a function describing temperature change over time, a positive gradient means the temperature is increasing.

Horizontal Tangent Lines

Criticality: 2

Tangent lines on a slope field that have a slope of zero, indicating points where the derivative dy/dx is equal to zero.

Example:

On a slope field, if you see a row of horizontal tangent lines along y=3, it suggests that y=3 might be an equilibrium solution or a line where local extrema occur.

Initial Condition

Criticality: 3

A specific value of the unknown function (or its derivative) at a particular point, used to determine the unique particular solution from a general solution.

Example:

To find the exact trajectory of a projectile, you need its starting height and initial velocity, which serve as initial conditions for the differential equations describing its motion.

Initial Condition

Criticality: 3

A specific value of the dependent variable at a given starting point (often $t=0$), used to determine the unique constant of integration in a differential equation's solution.

Example:

To predict the future temperature of a cooling object, you need its initial condition, such as its temperature at the moment it was removed from the oven.

Initial Conditions

Criticality: 3

Additional pieces of information provided to determine a specific solution from a family of solutions of a differential equation.

Example:

To find the exact path of a projectile, you need its starting position and velocity, which act as initial conditions for the differential equations describing its motion.

Integration Constant

Criticality: 3

An arbitrary constant, typically denoted by 'C', that is added to the result of an indefinite integral because the derivative of a constant is zero.

Example:

When integrating $\int x dx$ , the result is $\frac{x^2}{2} + C$ , where C is the Integration Constant representing any possible constant value.

Local Minimum/Maximum

Criticality: 2

Points on a function's graph where the function's value is either the smallest (minimum) or largest (maximum) within a specific neighborhood.

Example:

In a roller coaster's path, the lowest dip is a local minimum, and the highest peak is a local maximum.

Particular Solution

Criticality: 3

A specific function that satisfies a differential equation and also meets a given set of conditions, usually an initial or boundary condition.

Example:

If the general solution to dy/dx = 2x is y = x^2 + C, and we know y(0) = 5, then the particular solution is y = x^2 + 5.

Particular Solution

Criticality: 2

A specific solution to a differential equation obtained by using initial conditions to determine the value of the arbitrary constant(s) from the general solution.

Example:

If the General Solution is $y = Ce^x$ and $y(0)=2$ , then $2 = Ce^0 \Rightarrow C=2$ , leading to the Particular Solution $y = 2e^x$ .

Particular Solution

Criticality: 3

A unique solution derived from the general solution of a differential equation by applying specific initial or boundary conditions to determine the value of the arbitrary constant(s).

Example:

If the general solution is y = x² + C and the curve passes through (1, 3), then C = 2, making y = x² + 2 the particular solution.

Rate of change of a quantity is proportional to the size of the quantity

Criticality: 3

A fundamental principle stating that how quickly a quantity increases or decreases is directly dependent on its current amount.

Example:

Rates of change

Criticality: 2

How one quantity changes in relation to another, often expressed as a derivative. Differential equations are fundamentally used to model these relationships.

Example:

In physics, the rate of change of an object's position with respect to time is its velocity, and the rate of change of velocity is acceleration.

Regular Grid

Criticality: 1

A set of points arranged in a uniformly spaced pattern, typically used in slope fields to systematically draw tangent lines across the coordinate plane.

Example:

When constructing a slope field by hand, you often choose points like (0,0), (1,0), (0,1), etc., forming a regular grid to calculate slopes.

Separable Differential Equation

Criticality: 3

A type of first-order differential equation that can be written in the form $\frac{dy}{dx} = g(x)h(y)$, allowing the variables to be isolated on opposite sides of the equation.

Example:

The equation $\frac{dy}{dx} = \sin(x) \cdot e^y$ is a Separable Differential Equation because you can write it as $\frac{1}{e^y}dy = \sin(x)dx$ .

Separation of Variables

Criticality: 3

Example:

To solve $\frac{dy}{dx} = xy$ , you'd use Separation of Variables to get $\frac{1}{y}dy = xdx$ before integrating.

Separation of Variables

Criticality: 2

Example:

To solve dy/dx = y/x, you can use separation of variables to get dy/y = dx/x, which can then be integrated on both sides.

Separation of Variables

Criticality: 2

Example:

When solving $\frac{dy}{dx} = \frac{x^2}{y}$ , you would use separation of variables to rewrite it as $y , dy = x^2 , dx$ before integrating both sides.

Slope Field

Criticality: 3

A graphical representation of a differential equation, consisting of short tangent lines drawn at various points to illustrate the direction of solution curves.

Example:

When visualizing the flow of water in a river, a slope field could show the direction of the current at different locations.

Solution (to exponential growth & decay model)

Criticality: 3

The explicit function, $y = y_0 e^{kt}$, that satisfies the exponential growth and decay differential equation and its initial condition.

Example:

After solving the differential equation for a growing bacterial colony, the solution $P = P_0 e^{kt}$ allows you to predict the population size at any future time.

Tangent Line

Criticality: 3

A straight line that touches a curve at a single point, indicating the instantaneous direction or slope of the curve at that specific point.

Example:

If you trace the path of a thrown ball, a tangent line at any point on its trajectory would show the direction the ball is moving at that exact moment.

Verifying Solutions

Criticality: 2

The process of substituting a proposed solution and its derivatives back into the original differential equation to confirm that it satisfies the equation.

Example:

g(x)

Criticality: 2

In a separable differential equation of the form $\frac{dy}{dx} = g(x)h(y)$, $g(x)$ represents the function solely dependent on the variable $x$.

Example:

In the equation $\frac{dy}{dx} = x^2 \cos(y)$ , the g(x) term is $x^2$ .

h(y)

Criticality: 2

In a separable differential equation of the form $\frac{dy}{dx} = g(x)h(y)$, $h(y)$ represents the function solely dependent on the variable $y$.

Example:

In the equation $\frac{dy}{dx} = x^2 \cos(y)$ , the h(y) term is $\cos(y)$ .