Separable D.E.’s - Growth and Decay Applications Click here for examples.

In a Nut Shell: In general, differential equations where the independent variable, t,

does not appear explicitly such as:

dy/dt = f(y)

are termed "autonomous differential equations". Mathematical models representing

growth and decay fall into this category. The table below gives two of the simpler ones.

Differential equation involving “growth”

dy/dt = K y ; here rate of change of y is proportional to itself

subject to y(0) = y_o which represents the initial condition of y at t = 0

K = proportionality constant

Applications: a. population growth

b. annual rate (income)

dy/y = K dt (separate variables y and t and integrate)

ln y = K t + C_{1 ,} C₁ is the constant of integration

y = exp (Kt + C₁) = C e ^Kt

Differential equation involving “decay” (negative growth)

dy/dt = - K y ; here rate of change of y is proportional to its

subject to y(0) = y_o which represents the initial condition of y at t = 0

K = proportionality constant

i.e.

K = decay constant

K = sales decay constant

K = drug elimination constant

Applications: a. radioactive decay (radiocarbon dating)

b. annual rate (income)

dy/y = - K dt (separate variables y and t and integrate)

ln y = - K t + C_{1 ,} C₁ is the constant of integration

y = exp( ˗ Kt + C₁ ) = C e ^–Kt

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