Differential Equations

Unit Overview

A differential equation relates a function to its derivative(s). This unit is about modeling change and recovering a function from its rate. You will interpret and build models, visualize solution behavior with slope fields, verify solutions, and solve the one algebraic type AP AB requires — separable equations — including initial-value problems and the important applications of exponential growth/decay and logistic growth.

Scope note. AP Calculus AB covers separable first-order equations, slope fields, and growth/decay/logistic models. It does not require integrating factors, homogeneous or second-order equations (those are JEE / Class-12 topics). This study guide and its question bank are scoped to AP AB.

Big picture: Separate the variables, integrate both sides, then use an initial condition to pin down the constant.

Learning Objectives

• Interpret and build differential-equation models from a verbal description.
• Verify whether a given function solves a differential equation.
• Sketch and read slope fields; match a slope field to its equation.
• Solve separable equations and initial-value problems.
• Model and analyze exponential growth and decay (including half-life / doubling time).
• Understand the logistic model and its carrying capacity.

Prerequisite Concepts

• Antiderivatives and u-substitution (Unit 6).
• Properties of e^x and ln.
• Interpreting derivatives as rates (Unit 4).

Key Definitions

• Differential equation: an equation involving a function and its derivative(s).
• General solution: a family of solutions containing an arbitrary constant C.
• Particular solution: a specific solution determined by an initial condition.
• Slope field (direction field): a grid of short segments whose slopes equal dy/dx at each point, suggesting solution curves.
• Equilibrium solution: a constant solution where dy/dx = 0.
• Carrying capacity (M): the limiting value a logistic population approaches.

Important Models & Results

Exponential model: dy/dt = k y ⟹ y = y₀ e^(kt) (growth if k > 0, decay if k < 0).

• Half-life h: from e^(kh) = 1/2, k = −(ln 2)/h.
• Doubling time d: from e^(kd) = 2, k = (ln 2)/d.

Logistic model: dP/dt = k P (1 − P/M).

• Equilibria: P = 0 and P = M.
• Growth is fastest at P = M/2 (the solution curve's inflection point).
• As t → ∞, P → M.

Newton's Law of Cooling: dT/dt = k(T − A) (A = ambient temperature; equilibrium T = A).

Method: Separation of Variables

To solve dy/dx = g(x) h(y):

1. Separate: dy / h(y) = g(x) dx.
2. Integrate both sides (one + C suffices).
3. Solve for y if possible.
4. Apply the initial condition to find C (particular solution).

Concept Explanations

Modeling (7.1, 7.8). Translate phrases into equations: "rate proportional to amount" → dy/dt = ky; "proportional to the difference from A" → dy/dt = k(y − A); "rate proportional to amount and to room left" → logistic. Watch signs (decay/depreciation ⇒ k < 0).

Verifying solutions (7.2). Substitute the candidate (and its derivatives) into the equation and check it holds. y = Cx² solves x·y' = 2y for every C — the general solution.

Slope fields (7.3). At each point, plot a segment with slope dy/dx. If dy/dx depends only on x, segments in a vertical column are identical; only on y, identical along horizontal rows. Segments are horizontal where dy/dx = 0 (e.g., along y-axis for dy/dx = y). Solution curves flow along the segments.

Separation & IVPs (7.4–7.5). Most AP DE work is separable. After separating and integrating, use the initial point to determine C, then simplify. Example: dy/dx = x/y, y(1) = 1 → y² = x² + C → C = 0 → y = x (the branch through (1,1)).

Growth/decay & logistic (7.6–7.7). Recognize the exponential model instantly and use half-life/doubling relationships. For logistic, remember the population accelerates until M/2, then decelerates toward the carrying capacity.

Worked Examples

Example 1 (Separable). dy/dx = e^(x−y) → e^y dy = e^x dx → e^y = e^x + C.

Example 2 (Separable, harder). x(1 + y²)dx + y(1 + x²)dy = 0 → x/(1+x²)dx + y/(1+y²)dy = 0 → (1/2)ln(1+x²) + (1/2)ln(1+y²) = C₁ → (1 + x²)(1 + y²) = C. (from the Vidaara NCERT/JEE bank)

Example 3 (IVP). dy/dx = 2xy, y(0) = 3 → ln|y| = x² + C₁ → y = Ce^(x²), and y(0) = 3 gives y = 3e^(x²).

Example 4 (Half-life). A substance halves in 10 years: e^(−10k) = 1/2 → k = (ln 2)/10. (from the Vidaara NCERT/JEE bank)

Example 5 (Logistic reasoning). dP/dt = 0.5 P(1 − P/200) grows fastest at P = 100 (= M/2) and levels off near 200.

Example 6 (Slope field to solution). dy/dx = −x/y → y dy = −x dx → x² + y² = C: the solution curves are circles centered at the origin.

Graphical Interpretation

• A slope field is the "flow" of all solution curves; a particular solution threads through the field from its initial point.
• Exponential growth curves upward ever-faster; decay falls toward 0.
• The logistic curve is S-shaped (sigmoidal): concave up below M/2, concave down above, with a horizontal asymptote at M.

Common Mistakes

• Forgetting the constant C (and hence unable to apply the initial condition).
• Integrating without separating — variables must be fully separated first.
• Sign errors on decay (k must be negative) and on ∫ dy/y = ln|y|.
• Logistic mix-ups: fastest growth is at M/2, not at M or at the start.
• Trying non-AB methods (integrating factors, homogeneous substitutions) on the exam — AB expects separable solutions.

AP Exam Tips

• Show the separation and the integration step with + C on free-response — both earn points.
• After finding the general solution, apply the initial condition immediately to get the particular solution.
• For slope-field items, reason from where dy/dx is 0 or constant.
• Know the exponential solution y = y₀e^(kt) and half-life/doubling relations without re-deriving them.
• State logistic facts (equilibria 0 and M; fastest growth at M/2; limit M).

Unit Summary

• A differential equation models change; separation of variables is the AB solving method.
• General solution (with C) → apply initial condition → particular solution.
• Exponential: y = y₀e^(kt); use half-life/doubling to find k.
• Logistic: equilibria 0 and M, fastest growth at M/2, limit M.
• Slope fields visualize solution behavior without solving.

Next unit: using integrals for area, volume, average value, and motion.