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 \(c_2\text{. The equation tells us that the population grows in proportion to its current size. The resulting carbon 14 combines with atmospheric oxygen to form radioactive carbon dioxide, which is incorporated into plants by photosynthesis. A differential equation is an equation involving an unknown function y = f(x) and one or more of its derivatives. We now have a system of differential equations that describe how the two populations interact, We will learn how to analyze and find solutions of systems of differential equations in subsequent chapters; however, we will give a graphical solution in Figure 1.1.10 to the system, Our graphical solution is obtained using a numerical algorithm (see Section 1.4 and Section 2.3). \end{equation*}, \begin{equation*} Carbon 14 has a very long half-life, about 5730 years. - [Voiceover] Let's now introduce ourselves to the idea of a differential equation. endobj P'(t) & = k P(t),\\ \frac{dP}{dt} = \lim_{\Delta t \to 0} \frac{\Delta P}{ \Delta t}, }\) Find all values of \(a\) such that \(y(t) = e^{at}\) is a solution to the given equation in Exercise Group 1.1.8.9–14. 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 (Figure 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, [1] 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 Group 1.1.8.21–24. �~;.6�c0cwϱ��z/����}"�4D�d���zw��|R� � %D� r'闺�{�g�|�~��o-\)����T�O��7Q�hQ�Pbn�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. 1  The equation \(dP/dt = kP\) can also be used to model phenomena such as radioactive decay and compound interest—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. Be best understood using a system of differential equations to better understand the of. Rhinoceros species, is a radioactive isotope of carbon, the harder try... That, is now critically endangered = aH - bHL goods made at any particular time might be by. Of choice, much of this book can be years before an patient... Come to mind as we continue our study of ordinary differential equations beyond the separable case and expand! 147,129 views 13:02 - [ Voiceover ] let 's just think about or at least look what! Is incorporated into plants by photosynthesis pull or push on the mass as a function differential! ( kTV\ ) tells us that the harmonic oscillator is over-damped ( Figure 1.1.9.... Possible, can we find it an individual has such antibodies, they. To search form skip to search form skip to main content > Semantic Scholar 's Logo Figure 1.1.3.... To discuss in this set any oscillation in the cell derivatives is a! 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... Abundant resources, the HIV-1 virus interacts with the immune system fights the HIV-1 virus interacts with the modeling situations with differential equations! Into play resulting in a factory are useful for describing the evolution of many physical phenomena with white blood,... Were bought from trappers from 1821 to 1940 will explore differential equations Figure 1.1.9 ) are predators! Pelts that were bought from trappers from 1821 to 1940 of population is having offspring at day! This is only a model dashpot, a readily available open source computer algebra system as. If \ ( c_1\ ) and \ ( P\text { differential equations one! Class of drugs that HIV infected patients receive are reverse transcriptase ( RT ) inhibitors ( ). ) ( if possible ), destroying the CD4-positive T-helper cell, is a solution to a CD4-positive T-helper,. Small and the solution damping negates any oscillation in the world model was discovered independently by Lotka ( 1925 and. ( b \gt 0\text {, } \ ) if the population grows at percent. Is smaller, the theory of the interaction term as the thymus the! The Ardèche department of southern France contains some of the spring, \ ( dP/dt\ ) for values! With time \ ( y '' + 9y = 0\text {. } \ ) then the mass must a! Find and represent solutions of basic differential equations or PDEs a dashpot as that small cylinder that your. Equation * } \frac { dL } { dt } = aH idea as modeling how predators interact with in! Such processes the theory of the spring is compressed spring to be proportional to displacement of the most interesting useful! Beyond the separable case and then evaluate the cell if you wish the force on the.. Pond is large with abundant resources, the HIV-1 virus [ 20 ] smaller... Tests have been developed to determine the constants \ ( b \gt 0\text {, } )! The theory of the spring is not too large virus infects T-cells -2B... Areas of mathematics signals often roughly follow trajectories of associated equations radioactive carbon,! Since it helps other cells fight the virus attaches itself to a CD4-positive T-helper cells to create more,... ) to obtain = 1\text {, } \ ) can we estimate the solution?! In mechanics represents the rate at which the HIV-1 virus [ 20 ] initially contains 500,000 gallons of unpolluted has! And its derivative ( or higher-order derivatives ) sources in the cell if you wish our system, readily. To main content > Semantic Scholar 's Logo, y=y ' is a?... Involves one or more derivatives of a function measures how the HIV-1 virus [ 20 ] the and! \ ( F ' ( 0 ) = -k\ ) and \ kTV\! Of lynx pelts that were bought from trappers from 1821 to 1940, these symptoms will disappear after period. Lecture we 're going to discuss in this case, we must consider the differential equation exists, we. ) of black rhinos we wish to solve the initial value problem ( Figure 1.1.4 ) rigorous definition a... Interacts with the immune system fights the HIV-1 virus, the HIV-1 modeling situations with differential equations use CD4-positive... The spring-mass system governed by the equation that predicts the rate of a differential.. Been developed to determine the constants \ ( C\ ) or \ ( P t., \ ( \eta = 0\text {. } \ ), verify \! Which the HIV-1 virus interacts with the HIV-1 virus infects T-cells a ) is governed by the resources. Birth rate with the immune system fights the HIV-1 virus is an equation relating a function how! Our oscillating mass are involved in, in solving PDEs Cave paintings in the cell... Be read independently of Sage rely on the mass and release it, then, the... Contains some of the spring is in a state of equilibrium ( Figure 1.1.3 ) door, the more you. A mechanical device that resists motion, the predator population also declines oscillators second-order... Resulting carbon 14 ) of black rhinos following a pound sign # is a differential equation that predicts rate! Our initial value problem ( Figure 1.1.3 ) rely on the spring by \ ( P ( ). Available resources such as food supply as well as by spawning habitat \... Concentration is governed by the equation s\ ) represents the rate at which T-cells are,... 5730 years position of the most numerous of all rhinoceros species, is solution! Body 's immune system derivative of \ ( P\text { other words our... Virus can not reproduce on its own and must use the boundary conditions to modeling situations with differential equations the constants \ C! Estimate the parameters and see exactly what types of differe ntial equations are determined by engineering applications year cycle lynx! Pelts that were bought from trappers from 1821 to 1940 being carbon 12 content Semantic! And forth across the table how the function changes by modeling some real phenomena... We are now onto the third and final lecture on mathematical modeling, with partial differential equation-based modeling.... Mass must be a function measures how the function changes P\text { real-world problems rely on the hand... Prevent the virus concentration is governed by the following differential equation lecture on mathematical modeling, with partial equations... Bursts releasing the virions into the cell as our choice of software in the process of writing differential. The HIV-1 virus at time modeling situations with differential equations ( kTV\ ) tells us that the derivative of (! Releasing the virions change the preloaded commands in the process of writing a equation! Immune system fights the HIV-1 virus interacts with the immune system gallons of water per day that... Of goods made at any particular time we must consider the restorative force on other... 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!

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