Differential Equations in Context: Modeling and Slope Fields (AP Calc AB Unit 7)

Differential equation

An equation that relates an unknown function to one or more of its derivatives.

First-order differential equation

A differential equation in which the highest derivative that appears is the first derivative.

Rate of change (in modeling)

A description of how a quantity changes, typically written as a derivative such as dQ/dt or dy/dx.

Local behavior

The idea that a differential equation tells you the instantaneous slope at each point, not an explicit formula for the function all at once.

Independent variable

The input variable (often t for time or x for position) with respect to which the derivative is taken.

Initial condition

A starting value (e.g., y(0)=y0) used to pick out a specific solution from a family of solutions.

Initial value problem (IVP)

A differential equation together with an initial condition that specifies a unique solution (in typical AP settings).

Proportionality model

A model based on “proportional to” language, often leading to a differential equation like dy/dt = k·(expression).

Constant of proportionality (k)

The constant multiplier in a proportionality differential equation (e.g., dy/dt = ky) that controls growth/decay rate and direction.

Exponential growth/decay model

The model dy/dt = ky; k>0 gives growth and k<0 gives decay.

Limiting value (L) model

A model where the rate is proportional to the difference from a limiting value: dy/dt = k(L − y), often producing approach to equilibrium.

Equilibrium (in differential equations)

A value of y where the derivative is zero, so solutions can stay constant there.

Accumulation (“in minus out”) model

A structure for changing quantities: dQ/dt = (rate in) − (rate out).

Consistent units (in modeling)

The requirement that each term in a rate equation must have units of “amount of Q per unit time.”

Concentration

Amount per volume (e.g., kg/L); used to convert between “amount” and “rate” in mixing problems.

Flow rate

Volume per time (e.g., L/min); used with concentration to compute mass/amount rates in or out.

Mass (amount) rate

The rate of change of an amount (e.g., kg/min), often computed as (concentration)×(flow rate).

Autonomous differential equation

A differential equation that does not explicitly involve the independent variable (e.g., dy/dt = g(y)), so slopes depend only on y.

Slope field (direction field)

A graph showing short line segments whose slopes equal dy/dx = f(x,y) at many points, indicating how solution curves behave.

Solution curve

A smooth curve whose tangent slope at each point matches the slope field (i.e., matches the differential equation).

Stable equilibrium

An equilibrium where nearby solutions move toward the equilibrium over time.

Unstable equilibrium

An equilibrium where nearby solutions move away from the equilibrium over time.

Verifying a solution

Checking that a proposed function makes the differential equation true after substitution, and also satisfies any initial condition.

Satisfies a differential equation (on an interval)

After differentiating and substituting y=g(x), the equality g′(x)=f(x,g(x)) holds for all x in the interval (where defined).

Uniqueness / “no crossing” behavior

For well-behaved differential equations (typical in AP), two different solutions cannot pass through the same point, so solution curves should not intersect in a slope field.