Introduction to Differential Equations
Differential equations relate a function to its rate of change, forming the mathematical foundation of physics, engineering, biology, and economics.
What You Need to Know
Key Concept Diagram
A differential equation involves a function and one or more of its derivatives
dy/dx = f(x) means the rate of change of y with respect to x equals f(x)
Simple differential equations can be solved by integration
The general solution includes a constant C; an initial condition finds the particular solution
Exponential growth and decay are modelled by dy/dt = ky
Key Vocabulary
Differential equation
An equation that relates a function with its derivatives
Rate of change
How fast a quantity changes with respect to another variable
General solution
The family of solutions to a differential equation, including an arbitrary constant
Particular solution
A specific solution found by applying an initial or boundary condition
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
Solve dy/dx = 2x.
Question 2
The equation dy/dt = 3y models:
Question 3
If y = x^2 + C and y(0) = 5, what is the particular solution?
Key Concepts Summary
- ●A differential equation involves a function and one or more of its derivatives
- ●dy/dx = f(x) means the rate of change of y with respect to x equals f(x)
- ●Simple differential equations can be solved by integration
- ●The general solution includes a constant C; an initial condition finds the particular solution
- ●Exponential growth and decay are modelled by dy/dt = ky