Differential Equation
Definition
A differential equation is an equation that relates one or more derivatives of an unknown function with the function itself. It describes how the rate of change of a quantity depends on its current value.
Analogy
Imagine you are driving a car and your speedometer shows how fast you're going at any moment. A differential equation would be like having an equation that relates your acceleration (rate of change in speed) with your current speed.
Related terms
Ordinary Differential Equation (ODE): An ODE is a type of differential equation where only ordinary derivatives appear.
Partial Differential Equation (PDE): A PDE is a type of differential equation where partial derivatives appear, often used to describe physical phenomena involving multiple variables.
Initial Value Problem (IVP): An IVP involves solving a differential equation while also satisfying certain initial conditions, such as knowing the value or derivative at some specific point.
"Differential Equation" appears in:
Study guides (3)
Practice Questions (20+)
The amount of money in a savings account is given by the function A(t), where t is measured in years and A is measured in dollars. The rate of change of money is proportional to the square root of the current amount of money. When the account contains $10,000, the amount of money is increasing at $1,000/year. What is the differential equation that represents this situation?
The amount of radioactive material in a sample is given by the function N(t), where t is measured in years and N is measured in grams. The rate of change of the amount is proportional to the amount itself. If the initial quantity of material is 100 grams, and it begins decaying at a rate of 50 grams per year, what is the differential equation that represents the situation?
The acceleration of a racing car is proportional to the position of the car times the time passed. After 2 seconds, the car has traveled 12 meters and the acceleration is 6 meters per second per second. If x(t) is the position of the car, what is the differential equation that represents the situation?
The rate of change of the temperature of a cylindrical rod in Kelvin per second, T, is proportional to the radius of the cylinder in inches, r, and the height of the cylinder in inches, h. If the rate of change of the temperature is 60 Kelvin per second for a rod with radius 3 inches and height 2 inches, what is the differential equation that represents this situation?
The rate of change of the surface area of a sphere with respect to time is proportional to the square of the radius. At t = 0, the radius is 2 cm, and the rate of change of the surface area is 16 cm^2/s. If A(t) is the surface area of the sphere, what is the differential equation that represents this situation?
Which technique can be used to verify that a function is a solution to a differential equation?
For what value of k, will y = ksin(5x) + 2cos(4x) be a solution to the differential equation y'' + 16y = -81sin(5x)???
What is the main purpose of finding the max value of N when verifying a solution to a differential equation?
What should be done if the proposed solution satisfies the differential equation but fails to satisfy the initial conditions?
How are the solutions to a differential equation related to the slope field?
Find the particular solution to the differential equation y' = 2cos(x) with the initial condition y(0) = 3.
Given the differential equation y' = 4x - 6, find the solution that satisfies the initial condition y(2) = 5.
Given the differential equation y' = -2/x, find the particular solution that satisfies the initial condition y(1) = 2.
Consider the differential equation y' = 3x^2 - 2x. What is the solution to this differential equation?
How do families of functions differ in the context of solving a single differential equation?
Find the solution to the differential equation y' = 1/(x^2) with the initial condition y(1) = 1.
Given the differential equation y' = 2e^(2x), find the solution that satisfies the initial condition y(0) = 2.
Given the differential equation y' = 2x - y, find the particular solution that satisfies the initial condition y(0) = 1.
Given the differential equation dy/dx = x^2 + y and the initial condition y(1) = 3, identify the initial x, initial y, and slope values for the first step of Euler's method.
For the differential equation dy/dx - y = 1, which equation gives a value for the slope of the function y?
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