Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
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.
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