Differential Equations: every key term you need

Differential Equations

Equation with derivatives. dy/dx = f(x,y). Solution is function y = f(x). Models rates of change.

Initial Value Problems

Differential equation + initial condition. Example: dy/dx = 2x, y(0) = 1. Initial condition determines particular solution.

Slope Fields

Visual representation of dy/dx at each point. Sketched line segments. Solution curves follow field.

Separation of Variables

Rearrange: g(y)dy = f(x)dx. Integrate both sides: ∫g(y)dy = ∫f(x)dx + C.

Exponential Growth/Decay

dy/dx = ky. Solution: y = Ae^(kx). k>0: growth, k<0: decay. Half-life/doubling time common applications.

Logistic Growth

dy/dx = ky(1 - y/L). Models population with carrying capacity L. S-shaped curve.

Euler's Method

Numerical approximation of differential equations. y_(n+1) = yₙ + f(xₙ, yₙ)·Δx. Iterative process.

Particular vs General Solution

General: includes constant C (all solutions). Particular: specific solution from initial condition.

Homogeneous vs Nonhomogeneous

Homogeneous: dy/dx = f(y) (no explicit x term). Nonhomogeneous: dy/dx = f(x,y) (includes x).

Unit 8 Summary

Differential equations model rates of change. Separation of variables solves many. Exponential/logistic growth fundamental applications.

More AP Calculus BC guides

• Limits & Continuity
• Applications of Derivatives
• Parametric & Polar
• Infinite Series