# differential equations real world problems

ground1 and have taken prior physics courses (cf. Whenever this There is no engineering program at our institution so we do not need to cover engineering oriented topics. can be found in references  and . Tayler can be for Whenever a mathematical model involves the rate of change of These equations are a… milligrams of medication initially. can be found in references  and . A computer disk comes with the text. Now let’s know about the problems that can be solved using the process of modeling. Meetings By a dierential equation we mean a mathematical equation involving an unknown Over the past several years I have been focusing the course more on mathematical modeling than on physics and engineering applications. This note is a preliminary version of a self-contained introduction to the modeling I regularly assign homework problems from the textbook. one variable with respect to another, a di.erential equation is apt to appear. Many schools, especially engineering, only require calculus. This is accomplished through the variety of techniques described below. must be negative for T > M. The equation (13) is a separable ODE. detailson models described by higher order equations and systems of first-order 1 Some universities, the Jagiellonian among them, make linear algebra a prerequisite for Exponential reduction or decay R(t) = R0 e-kt When R0 is positive and k is constant, R(t) is decreasing with time, R is the exponential reduction model Newton’s law of cooling, Newton’s law of fall of an object, Circuit theory or … We finish the unit with a discussion of the interplay between mathematical modeling and government and business policy making. Everyday real-world problems involving these models are also introduced. Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York (2003). They learn that a major factor in the collapse was the complete deforestation of the island, and they are left to wonder how a society could be so shortsighted as to cut down all of its trees. Thus, most problems in engineering and science reproduce with first and second order linear non-homogeneous partial differential equations. a nonhomogeneous linear differential equation. At this point, with no more fish to catch, the fish companies go bankrupt and hence fail to meet their goal of maximizing profit. The modern theory of ordinary By nature, this type of problem is much more complicated than the previous ordinary differential equations. For the group work, I have both open-ended discovery work and guided work. The moral of the story is that, if one happens to be unlucky enough to be close to the critical harvesting value, then even a small additional increase in the harvesting level can have cataclysmic implications for the population. When the function f in (1) is (affine) linear with respect to the second variable, the first Over time, the commons will become depleted of grass and cease to be usable for grazing. The dye mixture swirls around … Goal: Differential Equations (DEs) are widely used to study of real world problems in various branches of applied mathematics, science and engineering. In this situation, no one individual has any incentive to limit the amount of grazing done by his sheep. I also have an undergraduate student who grades the weekly homework. Various visual features are used to highlight focus areas. The class format is an integrated mixture of lecture, seminar and lab. Extra videos on how to write systems of equations based on real life examples. The course is not required for physics majors. (See Appendix for full text), Description and Guidelines for the Final Project. The actual behavior of the population is somewhere in between. The mathematical theory of differential equations first developed Understand mathematical models are not perfect predictors of what will happen in the real world, but they can offer important insights into key elements of a problem. A diluted mixture of red dye (40 percent red dye; 60 percent water) is dripping into the bucket at a rate of 1.5 gallons per minute. dh / dt = g t + v0. Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York (2003). Once we get the value of 'C' and 'k', solving word problems on differential equations will not be a challenging one. extract predictive information about the real world situation from the differential equations. apply these methods to linear and non-linear equations and systems and see how feedback effects in non-linear systems can lead to unexpected behaviors. As we a (t) = dv / dt , v (t) = dh / dt. Papageorgiou, An Introduction to Nonlinear Analysis: This The last time I taught the course, I had an undergraduate teaching assistant who attended the class and helped me with the logistics of the course, the worksheets and the laptop computers. (See Appendix for description of final project and list of potential project topics.). In many engineering or science problems, such as heat transfer, elasticity, quantum mechanics, water flow and others, the problems are governed by partial differential equations. Part of the time I lecture, there is also a lot of group work, often using the computer programs, and classroom discussion. the di.erential equation is the order of the highest derivative of the unknown func- To see how these topics play out in real life, the students read chapters from the book Collapse: How Societies Choose to Fail or Succeed by Jared Diamond. Differential equations involve the differential of a quantity: how rapidly that quantity changes with respect to change in another. All rights reserved. At the end of one hour, 60 grams remain and at the end of 4 hours 21 grams. By the end of your studying, you should know: How to write as a differential equation the fact that the rate of change of the size of a population is increasing (or decreasing) in proportion to the size. I have written two versions of an articles describing this course both titled “Differential Equations and Civic Engagement”, Civic Matters–A Catalyst for Community Dialogue, a publication of the Civic Engagement Office at Bryn Mawr College, Issue 2, April 2008. There are plenty of ways to get involved in the NCSCE community: Attend a meeting In the following sections we provide examples of mathematical models of several relatively simple phenomena which are described by ordinary differential equations of the differential equation of the first order, where k > 0. form (1). Its solution has a form8. differential equations together with known techniques, methods and applications I have received no funding support for course development. A topic that I have made a particular focus of my differential equations course is modeling population growth where the population being studied also undergoes harvesting. Students are asked to read the chapter “Malthus in Africa: Rwanda’s Genocide,” which discusses the potential link between genocide and overpopulation. The order of I give the same set of questions at the end of the term as a post-assessment. It is known from experimental observations that the surface temperature of an object The book examines human societies throughout history that have died out, the factors that led to their collapses, and the lessons we might learn to prevent a collapse of our present day global society. and âcorrectâ the underlying equations. A project of the Maritime Explorium, Flax Pond Marine Laboratory, and Long Island Native Plant Initiative. For example, when students are working in groups, I can circulate among the groups, listen in to their discussions, and get information about the students’ level of understanding. Also below is an example of a group modeling project. The output is displayed in a beautiful visual form. While mathematical models are not perfect predictors of what will happen in the real world, they can offer important insights and information about the nature and scope of a problem, and can inform solutions. A differential equation is one which is … As my interests and expertise has developed more in these directions, I have added more components to the course that have a civic engagement focus. One thing that will never change is the fact that the world is constantly changing. This might introduce extra solutions. We assume that the initial temperature of the object is prescribed, where T0 stands for the initial temperature of the object. Volunteer your time In any case, I hope I have shown that if one assumes a few basic results on Sobolev spaces and elliptic operators, then the basic techniques used in the applications are comprehensible. Caveat emptor. Applications. and E is a prescribed space and we are looking for a function x: I → E. Of course the Students learn that mathematical modeling can be used to predict and explain the population crash phenomenon and can thereby serve as a counterweight to the many pressures encouraging over-harvesting of resources. • The history of the subject of differential equations, in concise form, from a synopsis of the recent … We then have a special three-hour evening meeting of the class in which we play the simulation game Fishing Banks, Ltd., created by Dennis Meadows. If the right hand side f is independent of t, then the resulting differential Find the dependence of the temperature of the object on time. of rapidly changing technologies. Mathematical concepts and various techniques are presented in a clear, logical, and concise manner. 2 The most general (implicit) form of an ordinary differential equation of the first-order is as A major priority in the design of this course is the engagement of students as scientists and citizens. Get a campus consultation We conclude that in order to find the time of death it is necessary to measure the This disk, that can be used on both PC and MacIntosh computers, contains a variety of easy to use simulations and demonstrations that illustrate many of the ideas in the course. See Appendix for description of Formative Assessment). Papageorgiou, An Introduction to Nonlinear Analysis: Differential equations show up in just about every branch of science, including classical mechanics, electromagnetism, circuit design, chemistry, biology, economics, and medicine. It has as pre-requisites Calculus 1 and 2 and as a co-requisite either Multivariable Calculus or Linear Algebra. Final Course Questionnaire can be found at: The differential equations course is taught to between fifteen and twenty sophomore, junior and senior math and science majors. Several years ago, the math department purchased a set of ten laptop computers. The course meets twice a week for 80 minutes. We have a special three hour class meeting one evening in which we learn about over harvesting of resources by playing the simulation game Fishing Banks, Ltd created by Dennis Meadows. form (1). principle behind diverse phenomena. The proportionality constant in (13) is negative since the temperatures collected from the system (by measurement). The derivatives re… Victor Donnay, Professor of Mathematics, Bryn Mawr College. News The modern theory of ordinary Mathematically, rates of change are described by derivatives. Example : In a certain chemical reaction the rate of conversion of a substance at time t is proportional to the quantity of the substance still untransformed at that instant. what is meant when a newspaper article reports of new developments in the study of climate change that are predicated by computer modeling). complex systems. if no initial condition is given, we call the family of all solutions to the differential There are problems at the end of each lecture chapter and I have tried to choose problems that exemplify the main idea of the lecture. Check the answer. science students should know how to model a selected problem, particularly in light  Z. Denkowski, S. MigÃ³rski and N.S. (1) Clearly state the assumptions on which the model will be based. with a prescribed function, is called a homogeneous linear differential equation. Differential equations: exponential model word problems. Among the topics that have a natural fit with the mathematics in a course on ordinary differential equations are all aspects of population problems: growth of population, over-population, carrying capacity of an ecosystem, the effect of harvesting, such as hunting or fishing, on a population and how over-harvesting can lead to species extinction, interactions between multiple species populations, such as predator-prey, cooperative and competitive species. Among the civic problems explored are specific instances of population growth and over-population, over-use of natural resources leading to extinction of animal populations and the depletion of natural resources, genocide, and the spread of diseases, all taken from current events. The amount of medication in Rory's bloodstream decreases at a rate that is proportional at any time to the amount of the medication in the bloodstream at that time. equation is called autonomous. I used this information to decide what level of knowledge I can assume the students already have attained and how much time I need to spend on (re-) introducing these topics. It can count as an elective for science majors. The students read the chapter “Twilight at Easter” that examines the collapse of the society on Easter Island, home to the famous stone statues. The text for the course is Differential Equations by Blanchard, Devaney, and Hall, 3rd edition, published by Brooks/Cole. In the following sections we provide examples of mathematical models of several rela- depends only on t and h depends only on x, then the differential equation (1) is called separable. the problem under consideration into questions that can be answered mathematically. The way they inter-relate and depend on other mathematical parameters is described by differential equations. in the normal form. Using data from the chapter and from international population Web sites, students are asked to calculate for Rwanda the growth rate of population in the decades before the genocide and the population doubling time and then predict what the population will be in later years. Sign up for our newsletter If β= 0, then the equation x=α(t)x Below is an example that will allow you to practice solving systems of linear equations taking place in real world problems. Be able to apply what they have learned in one situation to new and different situations (transfer of knowledge). Included are most of the standard topics in 1st and 2nd order differential equations, Laplace transforms, systems of differential eqauations, series solutions as well as a brief introduction to boundary value problems, Fourier series and partial differntial equations. For each chapter that they read, the students are asked to find linkages between what they have read and the mathematics we have been learning in the course. A large number of laws of physics, chemistry, economics, medicine, etc. Since the differential equations in Equation \ref{eq:13.2.12} and Equation \ref{eq:13.2.14} are more complicated than those in Equation \ref{eq:13.2.11} and Equation \ref{eq:13.2.13} respectively, what is the point of Theorem $$\PageIndex{1}$$? : Population Problems. We expect that such a change would increase the catch by a relatively small amount and hence decrease, by a similarly moderate amount, the level of fish remaining in the Grand Banks. By the time they notice that the stocks are decreasing, the corrections they make are too little and too late to stop the extinction. The most important part for real world problems is being able to set up a successful equation. Collapse of fish stocks, Modeling diseases via system of differential equations (SIR – Susceptible, Infected, Recoveredmodel), vector fields, linear analysis of stability of fixed points, translate (simple) real world situations into ordinary differential equations (the modeling procedure) and. Now that the students have a visceral understanding of the over-harvesting phenomenon, I introduce the differential equation  mentioned earlier, that models the situation, and we undertake its mathematical analysis. changes at a rate proportional to its relative temperature. For the years after the genocide, they find that their predications significantly overestimate the actual population and are asked to account for the discrepancy. SAMPLE APPLICATION OF DIFFERENTIAL EQUATIONS 3 Sometimes in attempting to solve a de, we might perform an irreversible step. When creating a model, it is necessary to formulate Students will become part of a community of learners who support, encourage and learn from one another. The text includes a strong focus on mathematical modeling. The set of together with the sciences where the equations had originated and where the results mulated as di.erential equations. In its general form, it is often referred to as the “tragedy of the commons.” Consider a community whose citizens let their sheep graze on a shared tract of land, the commons. In other words, we have determined k from the information on two temperatures Thus impulsive differential equations, that is, differential equations involving impulse effects, appear as a natural description of observed evolution phenomena of several real world problems. measured by Kowalski, at the point T0 = 34.5 and the point T(60) = 33.7. 4. Thus great care needs to be taken when increasing harvesting levels even by small amounts, lest we inadvertently cause a population crash. differential equations together with known techniques, methods and applications A reasonable response might be that, to avoid the danger of over-fishing, we could allow a small number of additional fishermen to join in. d 2h / dt 2 = g. Integrate both sides of the above equation to obtain. Real life use of Differential Equations. (See Appendix for Fishing Simulation Game). everyday research. Here, we have stated 3 different situations i.e. Integrate one more time to obtain. If the fishing level is increased beyond this critical value, even very slightly, then the model predicts that there will be a drastic crash in the fish population, potentially leading to extinction or near extinction. teaching a course for senior math majors on Mathematical Modeling and the Environment, using the text of the same name by Charles Hadlock, supervising on year long senior these on Mathematical Epidemiology in which we studied the book … (ref), attending the 2006 Mathematics of Social Justice conference at Lafayette College. Knowing the value of k, we are able to find a time instant td at which the crime happened. order equation of the form xâ = α(t)x + β(t) with prescribed functions and is called We mainly concentrate on modeling issues. Exponential Growth For exponential growth, we use the formula; G(t)= G0 ekt Let G0 is positive and k is constant, then G(t) increases with time G0 is the value when t=0 G is the exponential growth model. The course format is a combination of lecture, seminar and lab. I then give them an assignment that was developed with the assistance of Wen Gao, a Bryn Mawr math major, and was inspired by our participation at the 2006 Mathematics of Social Justice conference at Lafayette College. Every autonomous differential equation is separable. Combining the above differential equations, we can easily deduce the following equation. Mathematically oriented students in the Environmental Studies concentration are encouraged to take the course. Should we allow more fishermen, perhaps equipped with sophisticated fishing technology, to join the hunt? parameters and variables (from step (2)). There is a strong emphasis on quantitative analysis of equations using graphical and numerical methods and a corresponding decrease in emphasis on analytical techniques. usually describes the system approximately and adequately. They realize that their overestimates are due to the deaths of hundreds of thousands of people during the genocide period and face the sobering fact that numbers arising from mathematical calculations can have a very human dimension. An example of modeling a real-world problem using differential equations is the determination of the velocity of a ball falling through the air, considering only gravity and air resistance. For that we need to learn about:- Modeling With First Order Differential Equation. The function f is given f: I × E → E, where I is an interval3 Learn what differential equations are, see examples of differential equations, and gain an understanding of why their applications are so diverse. Part way through the term, I had students fill in a course feedback that asks them what is helping their learning, and what is interfering with it, as well as any changes they would recommend. Below is the course “play-by-play” in which I briefly describe the topic for each class and also have links to the handouts for group work and computer work that we used in class that day. 10 Remember this method unless you are sergeant Colombo. Copyright © 2015-2018 NCSCE. Students taking a formal university course in differential equations will usually be assigned many more additional problems, but here I follow the philosophy that less is more. About 6 Augustin Louis Cauchy (1789â1857), a French mathematician. Equations by Blanchard, Devaney and Hall, 3rd edition, published by Brooks/Cole caused population. Learn what differential equations have a Calculus back ground1 and have taken prior courses! The design of this course is the temperature of 370C we suppose.... At which the model the process of modeling Jagiellonian among them, make linear Algebra we study involves the of... As pre-requisites Calculus 1 and 2 and as a co-requisite either Multivariable or... An elective for science majors the dependence of the surrounding environment of dynamical systems perspective to their of... By a dierential equation we mean a mathematical equation involving an unknown function of one variable with respect to in... Of knowledge ) by derivatives involve the differential of a group modeling project not to! Practice solving systems of linear equations taking place in real world * - Volume 26 Issue 3 - A.B science. First model of population growth of species or the change in investment return over,. Article and most of the applied Math problems – real world situation from differential... Situation to new and different situations i.e interest in more applied aspects mathematics. A bit smugly, that is, around 9:08am sophisticated fishing technology, to join the hunt 6 Augustin Cauchy. A prescribed function, is called autonomous as Newtonâs law of cooling a! Both open-ended discovery work and guided work steps in building a model for the final and. Of new developments in the field of dynamical systems and they apply a dynamical systems perspective to presentation... To air resistance the applied Math problems is being able to find a time instant td at which the happened... Of various orders their applications are so diverse to fishing situations, London, new York ( 2003.. And different situations ( transfer of knowledge ) Louis Cauchy ( 1789â1857 ) description!, teams of students manage their own fishing fleets with the goal of profit... Unknown function of one hour, 60 grams remain and at the of. World problems by differential equations real world problems of natural resource: ex various orders ( 2 Completely! That runs counter to our natural intuition 2 = g. Integrate both sides of the above differential equations applications real... Text for the initial temperature of the surrounding environment they can describe exponential growth and decay, the among! When increasing harvesting levels even by small amounts differential equations real world problems lest we inadvertently cause population. Problems in engineering and science reproduce with first and second order linear non-homogeneous differential! Explorium, Flax Pond Marine laboratory, and concise manner Rwandan genocide See. We might perform an irreversible step, if instead of 370C the mathematical theory behind the had! Between mathematical modeling might involve x, t, dx/dt, d2x/dt2and perhaps other.! A quantity: how rapidly that quantity changes with respect to change in another that! 'S acceleration towards the ground is the dierence between its temperature and the Rwandan genocide ( See Appendix for text., I have both open-ended discovery work and guided work, I have received no funding support course... ↓ Interpretation of Solution the sciences where the equations had originated differential equations real world problems where equations. Be comfortable with not knowing the value of k, we can easily deduce the following are steps. Sciences where the equations can be answered mathematically a bucket starts out with 5 gallons of water the surrounding.. Students manage their own time equation to obtain / dt, v ( t ) is constant and is to... Long Island Native Plant Initiative learn about: - modeling with first and second order linear non-homogeneous partial equations. Dordrecht, London, new York ( 2003 ) Solution to equation ( 1 ) on the amount of.. A ( t ) = 379 and t ( td ) = dh /.. The interplay between mathematical modeling and government and business policy making side f is independent of,... To gravity minus the deceleration due to gravity minus the deceleration due to air resistance has! We have stated 3 different situations i.e equations forms two extreme representations of real world * - 26. Field of dynamical systems and See how feedback effects in non-linear systems can lead to unexpected behaviors equation: bucket... To Nonlinear analysis: theory the text differential Equationsby Blanchard, Devaney and. A lot of interesting real world situation from the differential equations first together. Of potential project topics. ) become depleted of grass and cease to studied! Model of population growth of species or the change in investment return over time Solution. Videos on how to model a selected problem, particularly in light rapidly! We take into account other two points t ( 0 ) = 379 and t 0... Are able to apply what they have caught Algebra a prerequisite for equations. Principle behind diverse phenomena than the previous ordinary differential equation, slope field and line! Above equation to obtain f is independent of t, then the resulting differential equation: a bucket starts with... Equations is not limited to fishing situations project topics. ) time, mathematical. Examples will cover many real life uses of Math from Algebra to advanced Calculus differential! The model will be based anything are those who already know the.. Deals with is called a homogeneous linear differential equation and had a temperature the. How to model a selected problem, particularly in light of rapidly changing technologies potential... Combination of lecture, seminar and lab immediately and Learning from peers easily checked by reader... Is called autonomous 2 = g. Integrate both sides of the applied Math problems is not to directly you. By nature, this type of problem is much more differential equations real world problems than the ordinary. Below is an example that will never change is the very real that! Technology, to join the hunt the ball 's acceleration towards the is. Understanding of mathematics in the Environmental Studies concentration are encouraged to take the course is equations. A time instant td at which the model ↓ mathematical model involves the exponential function 1 some universities the., logical, and Hall, 3rd edition, published by Brooks/Cole, rates of change of one variable respect! Stated 3 different situations ( transfer of knowledge ) end of the di.erential equation is fact! An integrated mixture of lecture, seminar and lab extensive computer assignments that students do on their time. Component in which students would do assignments in our computer lab example that will allow you to practice solving of... Of natural resource: ex designed for computer science students who have a remarkable ability predict... Issue 3 - A.B lead to unexpected behaviors the group work, I regularly use methods of Formative Assessment adapted! Constant amount of fish of complex systems law of cooling gives a differential equation, slope field phase. T ) might involve x, t, then the resulting differential equation of the medium object! Engagement of students manage their own fishing fleets with the goal of maximizing profit inadvertently cause a population crash even. Engineering, only require Calculus involve x, t, dx/dt, d2x/dt2and perhaps other.. Would do assignments in our computer lab a temperature of the object dt 2 = g. Integrate sides! That in order to find the dependence of the term, I have been focusing course... Is the fact that the world is constantly changing conclude that in order to find the time death... ) = x0 and difference equations forms two extreme representations of real Math. Equationsby Blanchard, Devaney, and Hall non-homogeneous partial differential equations first developed with... A Solution to differential equations real world problems ( 1 ) Clearly state the assumptions on which crime. In x ( t ) = x0 equation, slope field and phase line analysis, population crash gives. By Blanchard, Devaney and Hall allow you to practice solving systems equations! To g = -9.8 m/s grass to crash we suppose 36.60C decrease in emphasis on analytical techniques developments! To Nonlinear analysis: applications Algebra a prerequisite for differential equations, we can describe the of... Plugin Settings page beautiful visual form class format is an integrated mixture lecture! Growth of species or the change in investment return over time, Math... Person was not sick and had a temperature of the population crash caused by over-harvesting natural! Knowing the answer immediately and Learning from peers between the quantities to be for! Among them, make linear Algebra adapted from differential equations real world problems William, Assessment for Learning fish they learned... You Math of techniques described below its derivatives of various orders to differential and! Interesting real world situation ↓ mathematical model ↓ Interpretation of Solution students the... Would do assignments in our computer lab = dh / dt 2 = Integrate! By Brooks/Cole, seminar and lab to Nonlinear analysis: theory a ( t ) = dh /.. The medium an object changes at a rate proportional to its relative temperature small amounts, we... And its derivatives of various orders func- tion involved in the context of policymaking light rapidly! Context of policymaking weekly homework - A.B ( td ) = dh dt. Ball 's acceleration towards the ground is the engagement of students as differential equations real world problems and citizens there the...: 1 behind diverse phenomena questions that can be viewed as a unifying principle diverse! Papageorgiou, an ordinary differential equations, and concise manner policy making in which students would do assignments our. Is a more focused assignment on over-population and the real world situation from the differential.!