First-order models · Definition

Exponential growth and decay

differential equationsapplications

Before you start you need to know

When a population, a capital or a radioactive sample changes in proportion to what is already there, the simplest differential equation of modeling appears.

Definition

A quantity \(y\) with initial value \(y_0\) grows or decays exponentially when its rate of change is proportional to itself, with constant \(k\): it grows if \(k>0\), decays if \(k<0\).

\[y'=ky\;\Longrightarrow\; y(t)=y_0\,e^{kt}\]

The equation \(y'=ky\) is separable: \(\int dy/y=\int k\,dt\) gives \(\ln|y|=kt+c\), and with \(y(0)=y_0\) we get \(y(t)=y_0e^{kt}\). The sign of \(k\) decides everything: with \(k>0\) the quantity grows ever faster; with \(k<0\) it decays toward \(0\) without reaching it. Also, \(y=0\) is the only equilibrium.

Two quantities describe the pace without giving \(k\). The doubling time is \(t_d=\ln2/k\) when \(k>0\), and the half-life is \(t_{1/2}=\ln2/|k|\) when \(k<0\); neither depends on \(y_0\). That is why a radioactive sample loses half in each equal period, however much there is.

Guiding example: if a population of \(100\) doubles in \(2\) hours, then \(k=\ln2/2\) and after \(6\) hours there are \(100\cdot2^3=800\). To find \(k\) one more data point is enough, such as the value at another instant: use \(y(t_2)/y(t_1)=e^{k(t_2-t_1)}\) and solve.

The model is valid as long as the environment does not limit the process. No population grows forever; when resources run out a saturation term must be added, and that leads to the logistic model. Also avoid confusing it with linear growth: adding or subtracting a fixed amount per period is not exponential.

Think about it

If a sample loses \(10\,\%\) of its mass each year, what is \(e^{k}\)?

\(e^{k}=0.9\), so \(k=\ln0.9\approx-0.105\).

Exercises 10

  1. Solve \(y'=3y\) with \(y(0)=2\).
  2. Solve \(y'=-\tfrac12 y\) with \(y(0)=4\).
  3. A population satisfies \(y'=ky\), with \(y(0)=100\) and \(y(2)=200\). What is \(y(6)\)?
  4. A substance has a half-life of 5 years and satisfies \(y'=ky\). What is \(k\)?
  5. A capital grows with \(y'=0.04y\). After roughly how long does it double?
  6. In \(y'=ky\) with \(k<0\) and \(y(0)>0\), what happens to \(y(t)\) as \(t\to\infty\)?
  7. A 100 mg isotope keeps 80 mg after 3 years. With exponential decay, how many mg remain after 9 years?
  8. With \(y'=0.1y\), about when does the quantity triple?
  9. If \(y'=ky\) with \(y(1)=6\) and \(y(3)=24\), what is \(y(0)\)?
  10. A quantity doubles every 3 hours. Which equation does it satisfy?

What comes next

References

  • William E. Boyce, Richard C. DiPrima, Douglas B. Meade, Elementary Differential Equations and Boundary Value Problems, 12th edition, Wiley, 2021. §2.3
  • Dennis G. Zill, A First Course in Differential Equations with Modeling Applications, 11th edition, Cengage, 2018. §3.1