Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Looking for AP calculus AB differential equations? Then this AP Calculus AB lesson on Modeling Change with Separable Equations explains each idea step by step, with worked examples.
Quick answer: A differential equation relates a function to its derivative. Slope fields show the direction of solutions at each point, separation of variables solves equations of the form dy/dx = f(x)g(y), and exponential models solve dy/dt = ky, giving growth and decay with y = y₀e^(kt).
Standards: AP Calculus AB Unit 7 · Level: Grade 12 AP Calculus AB
What is a differential equation?
A differential equation relates a function to its own derivative, such as dydx = 2xy. A solution is a function that makes the equation true. A particular solution also meets an initial condition like y(0) = 3.
To verify a solution, differentiate it and substitute both the function and its derivative into the equation. For dydx = 3y, the function y = 5e3x works because y′ = 15e3x = 3(5e3x).
How do slope fields show solutions?
A slope field draws a short segment with slope dydx at each grid point. A solution curve follows the segments like a boat following a current. Where the formula gives 0, the segments are horizontal.
How does separation of variables work?
- Move every y to one side with dy and every x to the other with dx.
- Integrate both sides and include one constant C.
- Use the initial condition to find C, then solve for y if possible.
Solve dydx = 2xy with y(0) = 3. Separate: dy/y = 2x dx. Integrate: ln|y| = x2 + C. So y = Aex2. Since y(0) = 3, A = 3 and the solution is y = 3ex2.
What is exponential growth and decay?
If the rate of change is proportional to the amount, dydt = ky, then y = y0ekt. Growth has k > 0 and decay has k < 0. A half-life T gives k = −(ln 2)/T.
Forgetting the constant C at the moment of integration loses the initial condition. Add C immediately, before solving for y.
Verify every solution by differentiating it. It takes ten seconds and catches sign errors.
- A solution makes the differential equation true; a particular solution also matches an initial condition.
- A slope field draws the value of dy/dx as a short segment at each grid point.
- Separation of variables: move y to one side and x to the other, then integrate both.
- If dy/dt = ky, then y = y₀e^(kt).
- Always add the constant C when integrating, before using the initial condition.
Modeling Change with Separable Equations: frequently asked questions
How do I check that a function solves a differential equation?
Differentiate the function, then substitute the function and its derivative into the equation. If both sides become identical, the function is a solution.
What does a slope field tell me?
Each segment shows the slope of a solution passing through that point. You can sketch a solution by starting at the initial point and following the segments.
When can I use separation of variables?
Use it when the derivative can be written as a product of a function of x alone and a function of y alone. Then integrate each side separately.
How do I find the constant of proportionality from a half-life?
Set the amount equal to half of the start at the half-life time: ekT = 1/2, so k = −(ln 2)/T. A negative k indicates decay.
What is an equilibrium solution?
It is a constant solution where the derivative is zero, so the slope field has horizontal segments all along that line. For dy/dx = y(4 − y) the equilibria are y = 0 and y = 4.