Differential Equations
Lessons in this chapter
About Differential Equations
Differential Equations covers equations that relate a function to its own rate of change, and how to solve and interpret them.
Slope Fields visualizes a differential equation by sketching its rate of change at many points, showing the shape of its solutions before solving anything algebraically.
Finding Solutions covers the algebraic techniques for solving a differential equation exactly.
Modeling using Differential Equations closes the chapter by writing and solving differential equations for real situations, like population growth or cooling.
The three lessons, Slope Fields, Finding Solutions, and Modeling using Differential Equations, each open as a practice worksheet worked in the page. Problems are scored as you answer them, including the separating of variables, where a lost constant of integration changes the whole solution. Free to try, with no login needed for your first 3.
Frequently asked questions
Slope Fields draws the direction a solution must travel at each point, so the family of solutions can be seen before any one of them is found. Finding Solutions solves separable differential equations and uses an initial condition to pin down the one particular solution. Modeling using Differential Equations applies this to real situations such as exponential growth and decay. This is Unit 7 of the AP Calculus AB course framework, Differential Equations, and it goes past what the Common Core covers in high school.
A differential equation is an equation that relates a function to its own derivative (its own rate of change), rather than relating just input and output values directly.
A slope field sketches short line segments at many points, each showing the slope a solution curve would have there, giving a visual picture of every possible solution at once.
Common techniques include separating variables (grouping all the y-terms and dx/dy-terms) and then integrating both sides.
Real growth, decay, and cooling situations are modeled by writing a differential equation that says a quantity’s rate of change depends on the quantity itself.
Forgetting to include the constant of integration (and use given initial conditions to solve for it) when finding a differential equation’s particular solution.