Section 5.3: Differential Equations; Growth and Decay

Differential Equations

In general, a differential equation is an equation that involves an unknown function and one or more of its derivatives.

• The order of a differential equation is the highest derivative that it contains.
• A solution to a differential equation is a function f(x) which satisfies the equation when substituted for y.
• The general solution of a first-order differential equation satisfies the equation but involves a constant whose value is unknown. This is similar to the constant of integration seen in indefinite integrals.
• An initial condition specifies the value of a solution y when x = 0.
• A particular solution of a differential equation satisifes both the equation and an initial condition. These solutions will contain no unknown constants.

Slope Field

A slope field is a visual representation of a first-order differential equation.

• At points in the x-y plane, a short line segment is drawn with slope given by the value of dy/dx at that point.
• These line segments will be tangent to the graph of the particular solution to the differential equation that passes through these points.

Exponential Growth Law

If dQ/dt = rQ and Q(0) = Q₀, then Q = Q₀e^rt.

where

• Q₀ = amount of Q at t = 0
• r = relative growth rate (expressed as a decimal)
• t = time
• Q = quantity at time t

Limited Growth

This describes a situation where the rate of growth is proportional to the difference between the amount present and a fixed limit M:

dy/dt = k(M - y)

with solution given by:

y = M(1 - e^-kt)

Logistic Growth

This describes a situation where the rate of growth is proportional both to the amount present and to the difference between the amount present and a fixed limit M.

dy/dt = ky(M - y)

with solution given by:

y = M/(1 + ce^-kMt)