What You Need to Know
A differential equation is separable when you can rearrange it so that all of the terms are on one side and all of the terms are on the other.
The goal is to transform an equation involving into a form that can be integrated directly:
Once the variables are separated, integrate both sides. If an initial condition is provided, use it to determine the constant of integration.
The Basic Method
- Separate the variables. Put all expressions involving and on one side and all expressions involving on the other.
- Integrate both sides. Remember that both sides need an integration constant unless you use an equivalent single constant on one side.
- Use the initial condition, if one is given, to solve for the constant.
- Solve for the requested quantity. You may need to solve explicitly for or evaluate the solution at a particular time.
- Check the answer. Differentiate your solution and verify that it satisfies the original differential equation and initial condition.
Worked Examples
Solve the differential equation .
Find the particular solution to given .
A population satisfies . If the initial population is , find the population function.
A function satisfies with . Find .
Common Mistakes
When separating variables, is treated as a differential quantity. For example, from , multiplying by and by gives .
After integrating, include a constant. If you integrate both sides, you can write a constant on each side and then combine them into one arbitrary constant.
The cleanest approach is usually to separate and integrate first, then substitute the initial condition. This keeps the algebra organized and makes it easier to see exactly where the constant comes from.
An equation such as may be mathematically correct, but an AP problem may specifically ask for , a numerical value, or a rate. Read the question carefully and finish the algebra that it requests.
Practice Problems
Try each problem yourself before opening the solution. These problems progress from straightforward separation to initial-value and application problems.
Solve the differential equation .
Show answer
Find the particular solution to given .
Show answer
Find the particular solution to given .
Show answer
A population satisfies , where is measured in days. If , find .
Show answer
A function satisfies and . Find the value of .
Show answer
A quantity satisfies . If , find the value of .
Show answer
AP Exam Strategy
On AP Calculus free-response questions, separable differential equations are often only one part of a larger problem. You may need to solve the differential equation and then use the solution to answer a contextual question.
- Write the separation clearly. A correct setup earns you a much better chance of avoiding algebra mistakes later.
- Keep the constant. Do not lose the integration constant before applying the initial condition.
- Use the initial condition carefully. Substitute both coordinates from the ordered pair into the equation you obtained after integration.
- Watch the requested form. If the problem asks for a value such as , you may not need to completely solve for .
- Check units and interpretation. In a word problem, make sure your final answer actually answers the quantity the problem asks about.
The most important habit is simple: separate, integrate, apply the initial condition, and then answer the question. Once that sequence becomes automatic, many AP differential-equation problems become much more manageable.
Test yourself: AP Calculus practice exams
Ready to check what you know? Work an AP-style practice exam, then open the full solutions.
Want all 1,114 practice problems?
This tutorial covers one topic. Calc Pass covers all of AP Calculus AB & BC: full review, every practice question, and step-by-step solutions to each one.
Get the Calc Pass App
Scan to download and practice on the go, with instant scoring.
Vol. 1: 1,114 Questions
The complete practice question bank, organized by AB & BC topic.
View on Amazon