AP Calculus AB Unit 7 Notes: Solving Differential Equations by Separation

Differential equation

An equation that relates an unknown function (e.g., y as a function of x) to one or more of its derivatives (e.g., dy/dx).

Separation of variables

A method for solving certain first-order differential equations by rewriting so all y-terms (with dy) are on one side and all x-terms (with dx) are on the other, then integrating both sides.

Separable differential equation

A first-order differential equation that can be rearranged into the form dy/dx = g(x)h(y), allowing variables to be separated and integrated.

First-order differential equation

A differential equation involving only the first derivative of the unknown function (such as dy/dx) and no higher derivatives.

General solution

A family of solutions to a differential equation that includes an arbitrary constant (such as C), representing infinitely many solution curves.

Particular solution

A single specific solution obtained from the general solution by using an initial condition to determine the constant.

Initial condition

A given value of the solution at a specific input, such as y(x0)=y0 (or P(t0)=P0), used to select the correct solution curve.

Initial value problem (IVP)

A differential equation together with an initial condition; solving it produces a particular solution.

Implicit solution

A solution written as a relation between x and y (e.g., F(y)=G(x)+C) without explicitly isolating y.

Explicit solution

A solution written in the form y = (expression in x), with y isolated.

Equilibrium solution (constant solution)

A constant function solution (like y=0 or P=M) that makes the derivative zero; it can be missed if you divide by an expression that could be zero.

Dependent variable

The output variable that depends on the input (e.g., y or P), whose rate of change is described by the differential equation.

Independent variable

The input variable (e.g., x or t) with respect to which derivatives are taken.

Derivative notation (dy/dx, y′, dP/dt)

Different common notations for derivatives; dy/dx and y′ are equivalent, and dP/dt emphasizes the dependent variable P and independent variable t.

Differential form

A rewritten form of a differential equation (e.g., (1/h(y))dy = g(x)dx) used to integrate both sides during separation of variables.

Arbitrary constant (C)

A constant introduced after integration that accounts for the family of antiderivatives; only one combined constant is needed in a final general solution.

Combining constants

After integrating both sides, merging multiple constants (like C1 and C2) into a single constant to simplify and avoid errors.

Natural logarithm absolute value rule

The antiderivative ∫(1/y)dy = ln|y| + C; the absolute value is required in the general antiderivative.

Dropping absolute values (when justified)

Removing | | from expressions like ln|y| only after an initial condition or context guarantees y stays positive or negative on the interval considered.

Exponential growth/decay differential equation

A model where dy/dx is proportional to y (e.g., dy/dx = ky), leading to solutions of the form y = Ce^{kx}.

Logistic differential equation

A population model dP/dt = kP(1 − P/M) where growth is fast when P is small and slows as P approaches M.

Carrying capacity (M)

In the logistic model, the limiting population value that solutions tend toward as time increases (often as t → ∞).

Partial fractions (in logistic solving)

An algebra technique used to rewrite 1/[P(M−P)] as a sum of simpler fractions (e.g., (1/M)(1/P + 1/(M−P))) to integrate.

Separation pitfall: dividing by zero

The mistake of dividing by an expression involving the dependent variable (like y or P(M−P)) without checking values that make it zero, which can cause lost solutions.

Qualitative behavior from a differential equation

Interpreting how solutions behave (increasing/decreasing, leveling off near M, or becoming unbounded) based on the form of dy/dx, often without solving completely.