Revise later
SpaceTo flip
If confident
All Flashcards
Direct proportionality formula:
$a = kb$
Inverse proportionality formula:
$a = \frac{k}{b}$
Differential equation for 'rate of change of S w.r.t t is inversely proportional to x':
$\frac{dS}{dt} = \frac{k}{x}$
Differential equation for 'rate of change of A w.r.t t is proportional to the product of B and C':
$\frac{dA}{dt} = kBC$
What is the general form of a first-order differential equation?
$\frac{dy}{dx} = f(x, y)$
How do you represent 'y is proportional to x squared'?
$y = kx^2$
How do you represent 'z is inversely proportional to the square root of w'?
$z = \frac{k}{\sqrt{w}}$
What represents the general form for direct proportionality?
$\frac{y}{x} = k$
What represents the general form for inverse proportionality?
$xy = k$
Differential equation for 'The rate of change of P is proportional to P'?
$\frac{dP}{dt} = kP$
Explain proportionality in differential equations.
Proportionality describes how two quantities vary consistently with respect to each other, either directly or inversely, forming the basis of many differential equations.
How do differential equations model real-world scenarios?
They use rates of change to represent relationships between quantities, allowing us to understand and predict how these quantities change over time or with respect to other variables.
Why is the constant of proportionality 'k' important?
It determines the strength of the relationship between the variables in a proportional relationship, scaling the effect of one variable on another.
Explain how to translate a word problem into a differential equation.
Identify key phrases like 'rate of change,' 'proportional to,' or 'inversely proportional to.' Represent quantities with variables and translate the relationships into mathematical expressions involving derivatives.
Describe the significance of the derivative in a differential equation.
The derivative represents the rate of change of a function, providing information about how the function is changing at any given point.
What is the role of initial conditions in solving differential equations?
Initial conditions provide specific values of the function at a particular point, allowing us to find a unique solution to the differential equation.
Describe the difference between direct and inverse proportionality.
Direct proportionality means that as one quantity increases, the other increases proportionally. Inverse proportionality means that as one quantity increases, the other decreases proportionally.
What is the importance of units when modeling with differential equations?
Units ensure that the equation is dimensionally consistent and that the solution has the correct physical interpretation.
How can differential equations be used to model population growth?
By relating the rate of change of the population to the current population size, often incorporating factors like birth and death rates.
Explain how differential equations are used in physics.
They are used to describe motion, forces, and energy, allowing physicists to model and predict the behavior of physical systems.
What is a differential equation?
An equation involving derivatives, representing the relationship between a function and its rate of change.
What does $\frac{dy}{dx}$ represent?
The derivative of the function $y$ with respect to $x$, indicating the instantaneous rate of change of $y$ with respect to $x$.
Define 'directly proportional'.
If $a$ is directly proportional to $b$, then $a = kb$, where $k$ is a constant.
Define 'inversely proportional'.
If $a$ is inversely proportional to $b$, then $a = \frac{k}{b}$, where $k$ is a constant.
What is 'k' in differential equations?
Typically, $k$ represents the constant of proportionality.
What is the rate of change?
The rate at which a quantity is increasing or decreasing with respect to another quantity, often time.
What does modeling with differential equations mean?
Using differential equations to represent and analyze real-world scenarios involving rates of change.
What is a constant of proportionality?
The constant ($k$) in a proportionality equation that relates two variables.
What does the solution to a differential equation represent?
A function that satisfies the differential equation, describing the relationship between the variables.
Define independent variable.
A variable that is changed or controlled in a scientific experiment to test the effects on the dependent variable.