AP Calculus BC Unit 7 Notes: Solving Differential Equations (Separable, Exponential, Logistic)

Differential equation

An equation that relates an unknown function to one or more of its derivatives (e.g., dy/dx, dP/dt).

First-order differential equation

A differential equation involving only the first derivative of the unknown function (and not higher derivatives).

Separable differential equation

A first-order differential equation that can be written or rearranged as dy/dx = g(x)h(y), allowing x-terms and y-terms to be separated on opposite sides.

Separation of variables

A solving method where you algebraically rearrange so all y-expressions (with dy) are on one side and all x-expressions (with dx) are on the other, then integrate both sides.

General solution

A family of solutions to a differential equation that includes an arbitrary constant of integration (often C).

Particular solution

A single specific solution obtained by using an initial condition to determine the constant(s) in the general solution.

Initial condition

A given value of the function at a specific input (e.g., y(0)=5) used to find the constant and select a particular solution.

Constant of integration

The arbitrary constant (C) introduced when integrating, representing a family of antiderivatives/solutions.

Implicit solution

A solution not solved explicitly for the dependent variable (e.g., an equation relating x and y that may be left unsolved for y).

Leibniz notation

Derivative notation like dy/dx or dP/dt, commonly used in differential equations because it makes separation of variables visually natural.

Equilibrium (constant) solution

A constant function solution (e.g., y=c) that makes the derivative zero everywhere, so it satisfies the differential equation for all inputs.

Lost solution (from division)

A solution you may accidentally exclude by dividing by an expression that could be zero (e.g., dividing by y can lose the equilibrium solution y=0).

Exponential growth/decay differential equation

The model dy/dt = ky, expressing that the rate of change is proportional to the amount present.

Constant of proportionality (k)

The constant in dy/dt = ky; k>0 indicates growth, k<0 indicates decay, and its units are “per unit time.”

Exponential model solution form

The general solution to dy/dt = ky: y(t) = Ce^{kt} (often written y(t)=y0 e^{kt} after applying y(0)=y0).

Natural logarithm solution step

When separating and integrating 1/y, you get ln|y| (absolute value included): ∫(1/y)dy = ln|y| + C.

Doubling time

For y(t)=y0 e^{kt} with k>0, the time to double: Td = (ln 2)/k.

Half-life

For y(t)=y0 e^{kt} with k<0, the time to halve: T1/2 = ln(1/2)/k (a positive number because k is negative).

Logistic differential equation

A growth model with limiting behavior: dP/dt = kP(1 − P/L), where growth slows as P approaches L.

Carrying capacity (L)

In the logistic model, the long-term maximum sustainable value that solutions tend to approach (P approaches L).

Logistic slowdown factor

The term (1 − P/L) in dP/dt = kP(1 − P/L) that reduces growth as P gets close to L and makes growth negative if P > L.

Logistic equilibrium solutions

For dP/dt = kP(1 − P/L), the constant solutions P=0 and P=L obtained by setting the right-hand side equal to 0.

Partial fractions (logistic integration)

An algebra technique used to integrate 1/[P(L−P)] by rewriting it as (1/L)(1/P + 1/(L−P)).

Logistic solution (common form)

A standard explicit logistic solution: P(t)= L / (1 + Be^{−kt}), where B is determined from the initial condition.

Maximum logistic growth rate / inflection point

In the logistic model, the growth rate dP/dt is largest when P = L/2, which corresponds to the inflection point of the S-shaped solution curve.