Differential Equations: every key term you need (+ practice quiz)

More AP Calculus AB guides

• Limits and Continuity
• Derivatives
• Applications of Derivatives
• Integration

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.

Verifying a Solution

Substitute a proposed function and its derivatives into the differential equation; if both sides agree identically, the function is a solution.

General vs Particular Solution

The general solution carries an arbitrary constant; a particular solution uses an initial condition to fix that constant to a single number.

Separation Procedure

Move all y factors with dy and all x factors with dx, integrate both sides, add one constant, then solve for y and apply the initial condition.

Growth Model dy/dx = ky

The solutions are y = Ce^(kx); with dy/dx = 2xy the exponent instead becomes x², since the coefficient of y depends on x.

Doubling Time

For exponential growth y = y₀e^(kt), the doubling time is ln2/k, independent of the starting amount.

Newton's Law of Cooling

dT/dt = k(T - T_s) yields T = T_s + (T₀ - T_s)e^(kt) with k negative; the object's temperature approaches the surroundings asymptotically.

Equilibrium Solutions

Constant solutions occur where dy/dx = 0 for all x; for dy/dx = y - 2 the line y = 2 is an equilibrium that other solutions move away from.

Reading Slope Fields

Match a field to an equation by testing where slopes are zero, where they are constant along horizontal or vertical lines, and how they change sign.

Domain of a Particular Solution

A solution is valid only on the largest interval containing the initial x-value on which the solution is defined and differentiable.

Implicit Solutions

Sometimes integration leaves a relation such as y²/2 = sin x + C; solving for y requires choosing the branch that matches the initial condition's sign.

Area Between Curves

Integrate (top - bottom) with respect to x, or (right - left) with respect to y; intersection points supply the limits of integration.

Area Example

Between y = x and y = x² on [0,1], the region has area ∫(x - x²)dx = 1/2 - 1/3 = 1/6.

Disk Method

Revolving about a horizontal axis gives V = π∫[R(x)]²dx, where R is the distance from the axis to the curve.

Washer Method

With a gap between the region and the axis, V = π∫([R_outer]² - [R_inner]²)dx; the squares must be subtracted, never the radii first.

Revolving About a Vertical Axis

Rewrite the boundary as x = g(y) and integrate with respect to y, using the same disk or washer structure.

Known Cross Sections

For solids with cross sections perpendicular to an axis, V = ∫A(x)dx where A is the area formula for the given shape in terms of the base's width.

Square Cross Sections

If each cross section is a square with side equal to the distance between the bounding curves, A(x) = [top - bottom]².

Motion from Acceleration

Integrate a(t) to get v(t) with a constant fixed by v at a known time, then integrate again for position with a second constant.

Position from Velocity

s(b) = s(a) + ∫ from a to b of v(t)dt; the accumulated area supplies the change while the initial position supplies the offset.

Interpreting a Model in Context

An answer to an applied differential equation should include units and a sentence stating what the number means about the quantity being modeled.