A one-page visual summary of Differential Equations — slope fields, separation of variables, exponential and logistic models, and every exam trap, on a single screen.
What it covers: Modeling and verifying differential equations, slope fields, separation of variables, initial conditions, exponential models, and (BC) Euler’s method and logistic models.
Exam weight: About 6–12% of the AB exam and 6–9% of the BC exam.
The big question: How do we describe and solve situations defined by their rate of change?
Mathematical practices: Implementing Mathematical Processes, Connecting Representations, Justification, and Communication & Notation.
Key topics at a glance
Differential Equations
An equation relating a function to its derivatives (e.g., dy/dt = ky). A solution is a function that satisfies it.
Verifying Solutions
Differentiate the candidate and substitute into the equation; confirm both sides match.
Slope Fields
At each (x, y), draw a segment with slope dy/dx. Solution curves are tangent to the field.
Separation of Variables
Rewrite dy/dx = g(x)h(y) as (1/h) dy = g dx, integrate both sides (+C), then solve for y.
General vs. Particular
General solution has +C; a particular solution uses an initial condition to find C.
Exponential Models
dy/dt = ky → y = y₀eᵏᵗ. k > 0 growth, k < 0 decay; y₀ is the initial amount.
Euler’s Method (BC)
yₙ₊₁ = yₙ + f(xₙ, yₙ)·Δx — step along tangent lines to approximate a solution.
Logistic Models (BC)
dP/dt = kP(1 − P/L). Grows fastest at P = L/2; P → L (carrying capacity) as t → ∞.
The key terms and methods you must know
Differential equation — relates a function to its derivatives.
Solution — a function satisfying the equation.
General / particular solution — with +C / with C determined.
Initial condition — a point the solution must pass through.
Slope field — segments of slope dy/dx across the plane.
Separation of variables — (1/h)dy = g dx, then integrate.
Exponential model — dy/dt = ky → y = y₀eᵏᵗ.
Euler’s method (BC) — yₙ₊₁ = yₙ + f(xₙ,yₙ)Δx.
Logistic model (BC) — dP/dt = kP(1 − P/L).
Carrying capacity L (BC) — the limiting value P → L.
Key themes to remember
A differential equation describes change. Solving it recovers the function.
Slope fields visualize solutions without solving — follow the flow.
Separation, then integrate. The main analytic tool of the unit.
Initial condition → particular solution. It pins down the constant C.
“Proportional to the amount” means exponential. dy/dt = ky is the signal.
Common exam traps
Add + C when you integrate, and apply the initial condition to find it.
Not every equation is separable. dy/dx = x + y cannot be separated.
Solve for y explicitly when the question asks for y = f(x).
Euler’s method is only approximate — it uses tangent-line steps, not the exact curve.
Logistic growth is fastest at P = L/2, not at P = 0 or P = L.
Watch the sign of k: positive is growth, negative is decay.
Keep the constant of proportionality: dy/dt = ky, not dy/dt = y.