Differential Equations Basics: every key term you need

Differential Equations Basics

Equation involving function and derivatives; solution is function satisfying equation; types: separable, linear, etc.

Separable Differential Equations

dy/dx = f(x)g(y); separate variables: dy/g(y) = f(x)dx; integrate both sides; used in growth/decay problems.

Exponential Growth/Decay Model

dy/dt = ky (k>0: growth, k<0: decay); solution: y = y₀e^(kt); describes populations, radioactive decay, compound interest.

Logistic Growth Model

dy/dt = ky(1-y/L); carrying capacity L; S-shaped curve; solution: y = L/(1+Ae^(-kt)); models constrained growth.

Slope Fields

Visual representation of dy/dx at grid points; shows direction of solutions; solution curves tangent to slope segments.

Euler's Method

Numerical approximation: y_(n+1) = y_n + f(x_n, y_n)Δx; step-wise linear approximation of solution curve.

Initial Value Problems

Differential equation with initial condition y(x₀)=y₀; determines unique solution; fixes constant of integration.

Particular vs General Solutions

General: includes arbitrary constant C; particular: specific C value from initial condition.

L'Hôpital's Rule

For 0/0 or ∞/∞: lim f(x)/g(x) = lim f'(x)/g'(x); may apply rule multiple times; verify indeterminate form first.

Series and Sequences

Sequence: ordered list (a_n); series: sum of terms Σa_n; convergence: series approaches finite limit.

Geometric Series

Sum of r^n starting at r⁰; converges if |r|<1 to a/(1-r); diverges if |r|≥1.

Convergence Tests

nth-term test, integral test, comparison test, ratio test, root test; determine if infinite series converges.

Taylor Series

f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)²/2! + ...; polynomial approximation of function.

Maclaurin Series

Taylor series centered at a=0; e^x = 1+x+x²/2!+x³/3!+...; useful for approximations and integration.

Power Series

∑a_n(x-c)^n; converges for |x-c|<R (radius of convergence); represents function as infinite polynomial.

Radius of Convergence

R: distance from center where power series converges; test endpoints separately; found via ratio test.

Arc Length

L = ∫_a^b √(1+[f'(x)]²) dx; length of curve from (a,f(a)) to (b,f(b)); integrand comes from Pythagorean theorem.

Surface Area of Revolution

S = 2π∫_a^b f(x)√(1+[f'(x)]²) dx; area when rotating curve around x-axis; 2π·radius accounts for circular rotation.

Parametric Curves

x=f(t), y=g(t); traces curve as parameter varies; dy/dx = (dy/dt)/(dx/dt); allows smooth representation of complex curves.

Polar Coordinates

(r,θ): r is distance from origin, θ is angle; x=r·cos(θ), y=r·sin(θ); area: A = (1/2)∫_α^β r² dθ.

More AP Calculus AB guides

• Limits and Continuity
• Derivative Definition
• Extreme Value Theorem
• Antiderivative Definition