decay

Examples of decay in the following topics:

• Models Using Differential Equations

  • A good example of a physical system modeled with differential equations is radioactive decay in physics.
  • Over time, radioactive elements decay.
  • The mean lifetime, $\tau$ ("tau"), is the average lifetime of a radioactive particle before decay.
  • The decay constant, $\lambda$ ("lambda"), is the inverse of the mean lifetime.
  • For a number of radioactive particles $N$, the activity $A$, or number of decays per time is given by:

• Exponential Growth and Decay

  • Exponential decay occurs in the same way, providing the growth rate is negative.
  • If $\tau<0$ and $b > 1$, or $\tau > 0$ and $0 < b < 1$, then $x$ has exponential decay.
  • Apply the exponential growth and decay formulas to real world examples

• Derivatives of Exponential Functions

  • If a variable's growth or decay rate is proportional to its size—as is the case in unlimited population growth, continuously compounded interest, or radioactive decay—then the variable can be written as a constant times an exponential function of time.

• Logistic Equations and Population Grown

  • More quantitatively, as can be seen from the analytical solution, the logistic curve shows early exponential growth for negative $t$, which slows to linear growth of slope $\frac{1}{4}$ near $t = 0$, then approaches $y = 1$ with an exponentially decaying gap.

• Predator-Prey Systems

  • $\gamma y$ represents the loss rate of the predators due to either natural death or emigration; it leads to an exponential decay in the absence of prey.

• Exponential and Logarithmic Functions

  • The exponential function arises whenever a quantity grows or decays at a rate proportional to its current value.