AP Calculus: Differential Equations

This chapter introduces differential equations, which are equations that relate a function to its derivatives. The primary goal is to find the original function that satisfies the given equation. You will learn to distinguish between a general solution, which includes an arbitrary constant 'C', and a particular solution, where a specific value for 'C' is found using an initial condition.

The main problem-solving technique covered is separation of variables, where you algebraically rearrange an equation to group all terms of one variable on one side and all terms of the other variable on the other side before integrating. Other key topics include verifying a given function as a solution, and visualizing solutions graphically using slope fields. You will also learn to set up and solve differential equations that model real-world scenarios, particularly those involving rates of change like population growth and radioactive decay.

This chapter is a direct application of integration. Students who are not fluent in finding antiderivatives of various functions will find it very difficult, as integration is the central mechanical step. The conceptual challenge often comes from translating word problems into a mathematical equation and interpreting the meaning of a slope field.

Formulas for AP Calculus: Differential Equations

General First-Order Differential Equation: $\frac{dy}{dx} = f(x, y)$ The rate of change of y with respect to x is a function of x and/or y.
Separable Equation Form: $g(y) \, dy = f(x) \, dx$ The form required to solve by integrating both sides.
General Solution by Separation of Variables: $\int g(y) \, dy = \int f(x) \, dx + C$ Integrate both sides and add the constant of integration, C.
Exponential Growth and Decay Model: $\frac{dy}{dt} = ky$ The rate of change of a quantity y is directly proportional to the quantity itself.
Solution to Exponential Growth and Decay: $y(t) = y_0 e^{kt}$ y_0 is the initial value of y at time t=0, and k is the proportionality constant.
Newton's Law of Cooling: $\frac{dT}{dt} = k(T - T_s)$ T is the object's temperature and T_s is the constant surrounding temperature.
Logistic Differential Equation (BC Calculus): $\frac{dP}{dt} = kP\left(1 - \frac{P}{M}\right)$ Models growth with a maximum carrying capacity, M.