AP Calculus AB / Differential Equations
Lesson
Separating variables and naming the constant
When you can write g(y) dy = h(x) dx, integrate both sides and then use the initial point.
Learning goals
- Solve a separable DE with an initial condition.
- Recognize the exponential solutions of dy/dx = ky.
Explanation
Separation moves every y to one side and every x to the other, then integrates. Remember the +C on one side—or equivalently a constant after you combine. Solve for y when the prompt wants an explicit function. Check a domain if a logarithm or a denominator appears.
The equation y′ = ky says the rate is proportional to the amount. Solutions have the form y = y₀ e^{kt}. Growth when k > 0, decay when k < 0. This is a model, not a claim that every real population follows it forever.
Key terms
- Separable differential equation. A DE that can be written so each variable appears on only one side with its differential.
- Exponential growth or decay. A model in which the rate of change is proportional to the current value.
Common mistakes
- Dropping the absolute value or the constant and losing a family of solutions.
- Integrating both sides with respect to x while a leftover y is still in the wrong place.
Practice
Original Marlow Works items. Check the answer explanation after you try.
What is lim (x→3) of (x² − 9) / (x − 3)?
Original Marlow Works item — not a College Board question.
Take your time—this is practice, not a test.
Answer explanation
Why does y = 0 solve y′ = 3y even though you divided by y when separating?
Answer. The zero function has derivative zero, so it satisfies the DE; separation missed it because you divided by y.
Always glance back at lost constant solutions after dividing.
Related resources
External links with reuse status. Marlow Works is independent and does not copy restricted exam or textbook material.
Official / link only
AP Calculus AB course page
Official eight-unit framework and 2027 exam updates.
College Board · All rights reserved · accessed 2026-10-01
Open sourceReference only — do not copy
OpenStax College Algebra 2e
Algebra reference; current terms include noncommercial and AI-ingestion limits.
OpenStax · Restricted — see publisher page · accessed 2026-10-01
Open source