11.5: Summary
- Page ID
- 121143
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- A differential equation is a statement linking the rate of change of some state variable with current values of that variable. An example is the simplest population growth model: if \(N(t)\) is population size at time \(t\) :
\[\frac{d N}{d t}=k N \nonumber \]
- A solution to a differential equation is a function that satisfies the equation. For instance, the function \(N(t)=C e^{k t}\) (for any constant \(C\) ) is a solution to the unlimited population growth model (we check this by the appropriate differentiation). Graphs of such solutions (e.g. \(\mathrm{N}\) versus t) are called solution curves.
- To select a specific solution, more information (an initial condition) is needed. Given this information, e.g. \(N(0)=N_{0}\), we can fully characterize the desired solution.
- The decay equation is one representative of the same class of problems, and has an exponentially decaying solution.
\[\frac{d y}{d t}=-k y, \quad y(0)=y_{0} \quad \Rightarrow \quad \text { Solution: } y(t)=y_{0} e^{-k t} \nonumber \]
- So far, we have seen simple differential equations with simple (exponential) functions for their solutions. In general, it may be quite challenging to make the connection between the differential equation (stemming from some application or model) with the solution (which we want in order to understand and predict the behavior of the system.)
In this chapter, we saw examples in which a natural phenomenon (population growth, radioactive decay, cell growth) motivated a mathematical model that led to a differential equation. In both cases, that equation was derived by making a statement that tracked the amount or number or mass of a system over time. Numerous simplifications were made to derive each differential equation. For example, we assumed that the birth and mortality rates stay fixed even as the population grows to huge sizes.
With regard to a larger context.
- Our purpose was to illustrate how a simple model is created, and what such models can predict.
- In general, differential equation models are often based on physical laws (" \(F=m a ")\) or conservation statements ("rate in minus rate out equals net rate of change", or "total energy = constant").
- In biology, where the laws governing biochemical events are less formal, the models are often based on some mix of speculation and reasonable assumptions.
- In Figure \(11.8\) we illustrate how the scientific method leads to a cycle between the mathematical models and their test and validation using observations about the natural world.
- Identify each of the following with either exponential growth or exponential decay:
- \(y=20 e^{3 t}\)
- \(y=5 e^{-3 t}\)
- \(\frac{d y}{d t}=3 t\)
- \(\frac{d y}{d x}=-5 x\)
- Determine the doubling time of the exponential growth function \(N(t)=500 e^{2 t}\).
- Determine the half life of the of the exponential decay function \(N(t)=500 e^{-2 t}\).
- Consider the following figure depicting exponential growth:
What is the doubling time of this function?