where k is called the growth constant or the decay constant, as appropriate. Differential equations have aided the development of several fields of study. Derivatives of Algebraic Functions : Learn Formula and Proof using Solved Examples, Family of Lines with Important Properties, Types of Family of Lines, Factorials explained with Properties, Definition, Zero Factorial, Uses, Solved Examples, Sum of Arithmetic Progression Formula for nth term & Sum of n terms. If the body is heating, then the temperature of the body is increasing and gain heat energy from the surrounding and \(T < T_A\). L\ f 2 L3}d7x=)=au;\n]i) *HiY|) <8\CtIHjmqI6,-r"'lU%:cA;xDmI{ZXsA}Ld/I&YZL!$2`H.eGQ}. negative, the natural growth equation can also be written dy dt = ry where r = |k| is positive, in which case the solutions have the form y = y 0 e rt. (LogOut/ ( xRg -a*[0s&QM PRESENTED BY PRESENTED TO However, most differential equations cannot be solved explicitly. where k is a constant of proportionality. Ltd.: All rights reserved, Applications of Ordinary Differential Equations, Applications of Partial Differential Equations, Applications of Linear Differential Equations, Applications of Nonlinear Differential Equations, Applications of Homogeneous Differential Equations. %\f2E[ ^' Methods and Applications of Power Series By Jay A. Leavitt Power series in the past played a minor role in the numerical solutions of ordi-nary and partial differential equations. Ordinary differential equations are put to use in the real world for a variety of applications, including the calculation of the flow of electricity, the movement of an object like a pendulum, and the illustration of principles related to thermodynamics. Surprisingly, they are even present in large numbers in the human body. 4) In economics to find optimum investment strategies 8G'mu +M_vw@>,c8@+RqFh #:AAp+SvA8`r79C;S8sm.JVX&$.m6"1y]q_{kAvp&vYbw3>uHl etHjW(n?fotQT Bx1<0X29iMjIn7 7]s_OoU$l In geometrical applications, we can find the slope of a tangent, equation of tangent and normal, length of tangent and normal, and length of sub-tangent and sub-normal. They are defined by resistance, capacitance, and inductance and is generally considered lumped-parameter properties. Important topics including first and second order linear equations, initial value problems and qualitative theory are presented in separate chapters. APPLICATION OF DIFFERENTIAL EQUATIONS 31 NEWTON'S LAW OF O COOLING, states that the rate of change of the temperature of an object is proportional to the difference between its own temperature and th ambient temperature (i.e. When students can use their math skills to solve issues they could see again in a scientific or engineering course, they are more likely to acquire the material. Enjoy access to millions of ebooks, audiobooks, magazines, and more from Scribd. hbbd``b`:$+ H RqSA\g q,#CQ@ Chapter 7 First-Order Differential Equations - San Jose State University Instant PDF download; Readable on all devices; Own it forever; A few examples of quantities which are the rates of change with respect to some other quantity in our daily life . You can read the details below. Packs for both Applications students and Analysis students. 5) In physics to describe the motion of waves, pendulums or chaotic systems. What is the average distance between 2 points in arectangle? The Evolutionary Equation with a One-dimensional Phase Space6 . Application of Ordinary Differential equation in daily life - #Calculus by #Moein 8,667 views Mar 10, 2018 71 Dislike Share Save Moein Instructor 262 subscribers Click here for full courses and. There have been good reasons. Such kind of equations arise in the mathematical modeling of various physical phenomena, such as heat conduction in materials with mem-ory. Here "resource-rich" means, for example, that there is plenty of food, as well as space for, some examles and problerms for application of numerical methods in civil engineering. M for mass, P for population, T for temperature, and so forth. A 2008 SENCER Model. In this presentation, we tried to introduce differential equations and recognize its types and become more familiar with some of its applications in the real life. Hence, just like quadratic equations, even differential equations have a multitude of real-world applications. 3) In chemistry for modelling chemical reactions The graph above shows the predator population in blue and the prey population in red and is generated when the predator is both very aggressive (it will attack the prey very often) and also is very dependent on the prey (it cant get food from other sources). :dG )\UcJTA (|&XsIr S!Mo7)G/,!W7x%;Fa}S7n 7h}8{*^bW l' \ Already have an account? Numerical Solution of Diffusion Equation by Finite Difference Method, Iaetsd estimation of damping torque for small-signal, Exascale Computing for Autonomous Driving, APPLICATION OF NUMERICAL METHODS IN SMALL SIZE, Application of thermal error in machine tools based on Dynamic Bayesian Network. It relates the values of the function and its derivatives. Every home has wall clocks that continuously display the time. As is often said, nothing in excess is inherently desirable, and the same is true with bacteria. Reviews. 3) In chemistry for modelling chemical reactions If you read the wiki page on Gompertz functions [http://en.wikipedia.org/wiki/Gompertz_function] this might be a good starting point. ) The differential equation \({dP\over{T}}=kP(t)\), where P(t) denotes population at time t and k is a constant of proportionality that serves as a model for population growth and decay of insects, animals and human population at certain places and duration. Procedure for CBSE Compartment Exams 2022, Maths Expert Series : Part 2 Symmetry in Mathematics, Find out to know how your mom can be instrumental in your score improvement, 5 Easiest Chapters in Physics for IIT JEE, (First In India): , , , , NCERT Solutions for Class 7 Maths Chapter 9, Remote Teaching Strategies on Optimizing Learners Experience. As you can see this particular relationship generates a population boom and crash the predator rapidly eats the prey population, growing rapidly before it runs out of prey to eat and then it has no other food, thus dying off again. Hence, the order is \(1\). By solving this differential equation, we can determine the velocity of an object as a function of time, given its acceleration. Mathematics has grown increasingly lengthy hands in every core aspect. A differential equation is a mathematical statement containing one or more derivatives. A differential equation represents a relationship between the function and its derivatives. P,| a0Bx3|)r2DF(^x [.Aa-,J$B:PIpFZ.b38 Sign In, Create Your Free Account to Continue Reading, Copyright 2014-2021 Testbook Edu Solutions Pvt. Free access to premium services like Tuneln, Mubi and more. This introductory courses on (Ordinary) Differential Equations are mainly for the people, who need differential equations mostly for the practical use in their own fields. Differential equations have a remarkable ability to predict the world around us. Chemical bonds are forces that hold atoms together to make compounds or molecules. Chemical bonds include covalent, polar covalent, and ionic bonds. (iii)\)When \(x = 1,\,u(1,\,t) = {c_2}\,\sin \,p \cdot {e^{ {p^2}t}} = 0\)or \(\sin \,p = 0\)i.e., \(p = n\pi \).Therefore, \((iii)\)reduces to \(u(x,\,t) = {b_n}{e^{ {{(n\pi )}^2}t}}\sin \,n\pi x\)where \({b_n} = {c_2}\)Thus the general solution of \((i)\) is \(u(x,\,t) = \sum {{b_n}} {e^{ {{(n\pi )}^2}t}}\sin \,n\pi x\,. Thus, the study of differential equations is an integral part of applied math . Differential equations can be used to describe the relationship between velocity and acceleration, as well as other physical quantities. There are many forms that can be used to provide multiple forms of content, including sentence fragments, lists, and questions. By accepting, you agree to the updated privacy policy. 5) In physics to describe the motion of waves, pendulums or chaotic systems. Actually, l would like to try to collect some facts to write a term paper for URJ . The equations having functions of the same degree are called Homogeneous Differential Equations. Some of these can be solved (to get y = ..) simply by integrating, others require much more complex mathematics. What is Dyscalculia aka Number Dyslexia? We solve using the method of undetermined coefficients. What is an ordinary differential equation? Homogeneous Differential Equations are used in medicine, economics, aerospace, automobile as well as in the chemical industry. In describing the equation of motion of waves or a pendulum. Several problems in Engineering give rise to some well-known partial differential equations. Several problems in engineering give rise to partial differential equations like wave equations and the one-dimensional heat flow equation. Application of Partial Derivative in Engineering: In image processing edge detection algorithm is used which uses partial derivatives to improve edge detection. 300 IB Maths Exploration ideas, video tutorials and Exploration Guides, February 28, 2014 in Real life maths | Tags: differential equations, predator prey. Newtons Second Law of Motion states that If an object of mass m is moving with acceleration a and being acted on with force F then Newtons Second Law tells us. Where, \(k\)is the constant of proportionality. Applications of ordinary differential equations in daily life. \({d^y\over{dx^2}}+10{dy\over{dx}}+9y=0\). In mathematical terms, if P(t) denotes the total population at time t, then this assumption can be expressed as. written as y0 = 2y x. Q.4. Similarly, the applications of second-order DE are simple harmonic motion and systems of electrical circuits. It is often difficult to operate with power series. \(m{du^2\over{dt^2}}=F(t,v,{du\over{dt}})\). In the field of medical science to study the growth or spread of certain diseases in the human body. Then, Maxwell's system (in "strong" form) can be written: Application of differential equation in real life Dec. 02, 2016 42 likes 41,116 views Download Now Download to read offline Engineering It includes the maximum use of DE in real life Tanjil Hasan Follow Call Operator at MaCaffe Teddy Marketing Advertisement Advertisement Recommended Application of-differential-equation-in-real-life </quote> -(H\vrIB.)`?||7>9^G!GB;KMhUdeP)q7ffH^@UgFMZwmWCF>Em'{^0~1^Bq;6 JX>"[zzDrc*:ZV}+gSy eoP"8/rt: Application of differential equation in real life. By whitelisting SlideShare on your ad-blocker, you are supporting our community of content creators. Some other uses of differential equations include: 1) In medicine for modelling cancer growth or the spread of disease Problem: Initially 50 pounds of salt is dissolved in a large tank holding 300 gallons of water. 231 0 obj <>stream Important topics including first and second order linear equations, initial value problems and qualitative theory are presented in separate chapters. Hi Friends,In this video, we will explore some of the most important real life applications of Differential Equations.Time Stamps-Introduction-0:00Population. Also, in medical terms, they are used to check the growth of diseases in graphical representation. Thus \({dT\over{t}}\) < 0. Leading AI Powered Learning Solution Provider, Fixing Students Behaviour With Data Analytics, Leveraging Intelligence To Deliver Results, Exciting AI Platform, Personalizing Education, Disruptor Award For Maximum Business Impact, Copyright 2023, Embibe. The differential equation is the concept of Mathematics. If a quantity y is a function of time t and is directly proportional to its rate of change (y'), then we can express the simplest differential equation of growth or decay. 40 Thought-provoking Albert Einstein Quotes On Knowledge And Intelligence, Free and Appropriate Public Education (FAPE) Checklist [PDF Included], Everything You Need To Know About Problem-Based Learning. A differential equation is an equation that relates one or more functions and their derivatives. How many types of differential equations are there?Ans: There are 6 types of differential equations. %PDF-1.6 % The second-order differential equation has derivatives equal to the number of elements storing energy. Then we have \(T >T_A\). For a few, exams are a terrifying ordeal. The constant k is called the rate constant or growth constant, and has units of inverse time (number per second). 7 Manipulatives For Learning Area And Perimeter Concepts, Skimming And Scanning: Examples & Effective Strategies, 10 Online Math Vocabulary Games For Middle School Students, 10 Fun Inference Activities For Middle School Students, 10 Effective Reading Comprehension Activities For Adults, NumberDyslexia is a participant in the Amazon Services LLC Associates Program, an affiliate advertising program designed to provide a means for sites to earn advertising fees by advertising and linking to Amazon.com. HUmk0_OCX- 1QM]]Nbw#`\^MH/(:\"avt Growth and Decay. One of the earliest attempts to model human population growth by means of mathematics was by the English economist Thomas Malthus in 1798. They can be used to model a wide range of phenomena in the real world, such as the spread of diseases, the movement of celestial bodies, and the flow of fluids. Some make us healthy, while others make us sick. To demonstrate that the Wronskian either vanishes for all values of x or it is never equal to zero, if the y i(x) are solutions to an nth order ordinary linear dierential equa-tion, we shall derive a formula for the Wronskian. In the case where k is k 0 t y y e kt k 0 t y y e kt Figure 1: Exponential growth and decay. Do not sell or share my personal information. In recent years, there has been subject so far-reaching of research in derivative and differential equation because of its performance in numerous branches of pure and applied mathematics. Game Theory andEvolution. ) ]JGaGiXp0zg6AYS}k@0h,(hB12PaT#Er#+3TOa9%(R*%= by MA Endale 2015 - on solving separable , Linear first order differential equations, solution methods and the role of these equations in modeling real-life problems. In the description of various exponential growths and decays. \(ln{|T T_A|}=kt+c_1\) where c_1 is a constant, Hence \( T(t)= T_A+ c_2e^{kt}\) where c_2 is a constant, When the ambient temperature T_A is constant the solution of this differential equation is. So, for falling objects the rate of change of velocity is constant. To learn more, view ourPrivacy Policy. We regularly post articles on the topic to assist students and adults struggling with their day to day lives due to these learning disabilities. We've updated our privacy policy. Numerical Methods in Mechanical Engineering - Final Project, A NEW PARALLEL ALGORITHM FOR COMPUTING MINIMUM SPANNING TREE, Application of Derivative Class 12th Best Project by Shubham prasad, Application of interpolation and finite difference, Application of Numerical Methods (Finite Difference) in Heat Transfer, Some Engg. It involves the derivative of a function or a dependent variable with respect to an independent variable. This book presents the application and includes problems in chemistry, biology, economics, mechanics, and electric circuits. 4.7 (1,283 ratings) |. But how do they function? Since many real-world applications employ differential equations as mathematical models, a course on ordinary differential equations works rather well to put this constructing the bridge idea into practice. Differential Equations have already been proved a significant part of Applied and Pure Mathematics. Roughly speaking, an ordinary di erential equation (ODE) is an equation involving a func- Numerical case studies for civil enginering, Essential Mathematics and Statistics for Science Second Edition, Ecuaciones_diferenciales_con_aplicaciones_de_modelado_9TH ENG.pdf, [English Version]Ecuaciones diferenciales, INFINITE SERIES AND DIFFERENTIAL EQUATIONS, Coleo Schaum Bronson - Equaes Diferenciais, Differential Equations with Modelling Applications, First Course in Differntial Equations 9th Edition, FIRST-ORDER DIFFERENTIAL EQUATIONS Solutions, Slope Fields, and Picard's Theorem General First-Order Differential Equations and Solutions, DIFFERENTIAL_EQUATIONS_WITH_BOUNDARY-VALUE_PROBLEMS_7th_.pdf, Differential equations with modeling applications, [English Version]Ecuaciones diferenciales - Zill 9ed, [Dennis.G.Zill] A.First.Course.in.Differential.Equations.9th.Ed, Schaum's Outline of Differential Equations - 3Ed, Sears Zemansky Fsica Universitaria 12rdicin Solucionario, 1401093760.9019First Course in Differntial Equations 9th Edition(1) (1).pdf, Differential Equations Notes and Exercises, Schaum's Outline of Differential Equation 2ndEd.pdf, [Amos_Gilat,_2014]_MATLAB_An_Introduction_with_Ap(BookFi).pdf, A First Course in Differential Equations 9th.pdf, A FIRST COURSE IN DIFFERENTIAL EQUATIONS with Modeling Applications. Such a multivariable function can consist of several dependent and independent variables. Bernoullis principle can be applied to various types of fluid flow, resulting in various forms of Bernoullis equation. Partial Differential Equations and Applications (PDEA) offers a single platform for all PDE-based research, bridging the areas of Mathematical Analysis, Computational Mathematics and applications of Mathematics in the Sciences. to the nth order ordinary linear dierential equation. the temperature of its surroundi g 32 Applications on Newton' Law of Cooling: Investigations. 'l]Ic], a!sIW@y=3nCZ|pUv*mRYj,;8S'5&ZkOw|F6~yvp3+fJzL>{r1"a}syjZ&. Differential Equations are of the following types. What are the applications of differential equations?Ans:Differential equations have many applications, such as geometrical application, physical application. The applications of partial differential equations are as follows: A Partial differential equation (or PDE) relates the partial derivatives of an unknown multivariable function. Answer (1 of 45): It is impossible to discuss differential equations, before reminding, in a few words, what are functions and what are their derivatives. (LogOut/ With a step-by-step approach to solving ordinary differential equations (ODEs), Differential Equation Analysis in Biomedical Science and Engineering: Ordinary Differential Equation Applications with R successfully applies computational techniques for solving real-world ODE problems that are found in a variety of fields, including chemistry, The major applications are as listed below. Since velocity is the time derivative of the position, and acceleration is the time derivative of the velocity, acceleration is the second time derivative of the position. which can be applied to many phenomena in science and engineering including the decay in radioactivity. Ive also made 17 full investigation questions which are also excellent starting points for explorations. APPLICATION OF HIGHER ORDER DIFFERENTIAL EQUATIONS 1. For example, the use of the derivatives is helpful to compute the level of output at which the total revenue is the highest, the profit is the highest and (or) the lowest, marginal costs and average costs are the smallest. But then the predators will have less to eat and start to die out, which allows more prey to survive. Newtons empirical law of cooling states that the rate at which a body cools is proportional to the difference between the temperature of the body and that of the temperature of the surrounding medium, the so-called ambient temperature. All rights reserved, Application of Differential Equations: Definition, Types, Examples, All About Application of Differential Equations: Definition, Types, Examples, JEE Advanced Previous Year Question Papers, SSC CGL Tier-I Previous Year Question Papers, SSC GD Constable Previous Year Question Papers, ESIC Stenographer Previous Year Question Papers, RRB NTPC CBT 2 Previous Year Question Papers, UP Police Constable Previous Year Question Papers, SSC CGL Tier 2 Previous Year Question Papers, CISF Head Constable Previous Year Question Papers, UGC NET Paper 1 Previous Year Question Papers, RRB NTPC CBT 1 Previous Year Question Papers, Rajasthan Police Constable Previous Year Question Papers, Rajasthan Patwari Previous Year Question Papers, SBI Apprentice Previous Year Question Papers, RBI Assistant Previous Year Question Papers, CTET Paper 1 Previous Year Question Papers, COMEDK UGET Previous Year Question Papers, MPTET Middle School Previous Year Question Papers, MPTET Primary School Previous Year Question Papers, BCA ENTRANCE Previous Year Question Papers, Study the movement of an object like a pendulum, Graphical representations of the development of diseases, If \(f(x) = 0\), then the equation becomes a, If \(f(x) \ne 0\), then the equation becomes a, To solve boundary value problems using the method of separation of variables. If after two years the population has doubled, and after three years the population is \(20,000\), estimate the number of people currently living in the country.Ans:Let \(N\)denote the number of people living in the country at any time \(t\), and let \({N_0}\)denote the number of people initially living in the country.\(\frac{{dN}}{{dt}}\), the time rate of change of population is proportional to the present population.Then \(\frac{{dN}}{{dt}} = kN\), or \(\frac{{dN}}{{dt}} kN = 0\), where \(k\)is the constant of proportionality.\(\frac{{dN}}{{dt}} kN = 0\)which has the solution \(N = c{e^{kt}}. The. Newtons law of cooling can be formulated as, \(\frac{{dT}}{{dt}} = k\left( {T {T_m}} \right)\), \( \Rightarrow \frac{{dT}}{{dt}} + kT = k{T_m}\). For example, if k = 3/hour, it means that each individual bacteria cell has an average of 3 offspring per hour (not counting grandchildren). In medicine for modelling cancer growth or the spread of disease In the calculation of optimum investment strategies to assist the economists. The second-order differential equations are used to express them. Have you ever observed a pendulum that swings back and forth constantly without pausing? Firstly, l say that I would like to thank you. During the past three decades, the development of nonlinear analysis, dynamical systems and their applications to science and engineering has stimulated renewed enthusiasm for the theory of Ordinary Differential Equations (ODE). How understanding mathematics helps us understand human behaviour, 1) Exploration Guidesand Paper 3 Resources. If we integrate both sides of this differential equation Z (3y2 5)dy = Z (4 2x)dx we get y3 5y = 4x x2 +C. I was thinking of modelling traffic flow using differential equations, are there anything specific resources that you would recommend to help me understand this better? 40K Students Enrolled. The picture above is taken from an online predator-prey simulator . If you are an IB teacher this could save you 200+ hours of preparation time. The differential equation of the same type determines a circuit consisting of an inductance L or capacitor C and resistor R with current and voltage variables. \(p(0)=p_o\), and k are called the growth or the decay constant. systems that change in time according to some fixed rule. if k>0, then the population grows and continues to expand to infinity, that is. When \(N_0\) is positive and k is constant, N(t) decreases as the time decreases. Nonhomogeneous Differential Equations are equations having varying degrees of terms. hZqZ$[ |Yl+N"5w2*QRZ#MJ 5Yd`3V D;) r#a@ We assume the body is cooling, then the temperature of the body is decreasing and losing heat energy to the surrounding. Applied mathematics involves the relationships between mathematics and its applications. Partial Differential Equations are used to mathematically formulate, and thus aid the solution of, physical and other problems involving functions of several variables, such as the propagation of heat or sound, fluid flow, elasticity, electrostatics, electrodynamics, thermodynamics, etc. e - `S#eXm030u2e0egd8pZw-(@{81"LiFp'30 e40 H! 100 0 obj <>/Filter/FlateDecode/ID[<5908EFD43C3AD74E94885C6CC60FD88D>]/Index[82 34]/Info 81 0 R/Length 88/Prev 152651/Root 83 0 R/Size 116/Type/XRef/W[1 2 1]>>stream The Integral Curves of a Direction Field4 . Differential equations have applications in various fields of Science like Physics (dynamics, thermodynamics, heat, fluid mechanics, and electromagnetism), Chemistry (rate of chemical reactions, physical chemistry, and radioactive decay), Biology (growth rates of bacteria, plants and other organisms) and Economics (economic growth rate, and population growth rate). They are used in a wide variety of disciplines, from biology This requires that the sum of kinetic energy, potential energy and internal energy remains constant. So, with all these things in mind Newtons Second Law can now be written as a differential equation in terms of either the velocity, v, or the position, u, of the object as follows. %%EOF %%EOF An ordinary differential equation is a differential equation containing one or more functions of one independent variable and the derivatives of those functions. Recording the population growth rate is necessary since populations are growing worldwide daily. They can get some credit for describing what their intuition tells them should be the solution if they are sure in their model and get an answer that just does not make sense. Q.1. 0 Im interested in looking into and potentially writing about the modelling of cancer growth mentioned towards the end of the post, do you know of any good sources of information for this? In all sorts of applications: automotive, aeronautics, robotics, etc., we'll find electrical actuators. di erential equations can often be proved to characterize the conditional expected values. From an educational perspective, these mathematical models are also realistic applications of ordinary differential equations (ODEs) hence the proposal that these models should be added to ODE textbooks as flexible and vivid examples to illustrate and study differential equations. Can Artificial Intelligence (Chat GPT) get a 7 on an SL Mathspaper? The differential equation is regarded as conventional when its second order, reflects the derivatives involved and is equal to the number of energy-storing components used. For example, as predators increase then prey decrease as more get eaten. {dv\over{dt}}=g. The solution of this separable firstorder equation is where x o denotes the amount of substance present at time t = 0. If you want to learn more, you can read about how to solve them here. Consider the differential equation given by, This equation is linear if n=0 , and has separable variables if n=1,Thus, in the following, development, assume that n0 and n1. The differential equation for the simple harmonic function is given by. More complicated differential equations can be used to model the relationship between predators and prey. The three most commonly modelled systems are: In order to illustrate the use of differential equations with regard to population problems, we consider the easiest mathematical model offered to govern the population dynamics of a certain species. Functions 6 5. Adding ingredients to a recipe.e.g. Almost all of the known laws of physics and chemistry are actually differential equations , and differential equation models are used extensively in biology to study bio-A mathematical model is a description of a real-world system using mathematical language and ideas. The absolute necessity is lighted in the dark and fans in the heat, along with some entertainment options like television and a cellphone charger, to mention a few. Graphic representations of disease development are another common usage for them in medical terminology. The general solution is or written another way Hence it is a superposition of two cosine waves at different frequencies. The order of a differential equation is defined to be that of the highest order derivative it contains. Ordinary Differential Equations with Applications Authors: Carmen Chicone 0; Carmen Chicone. The following examples illustrate several instances in science where exponential growth or decay is relevant. With such ability to describe the real world, being able to solve differential equations is an important skill for mathematicians. Research into students thinking and reasoning is producing fresh insights into establishing and maintaining learning settings where students may develop a profound comprehension of mathematical ideas and procedures, in addition to novel pedagogical tactics. Solution of the equation will provide population at any future time t. This simple model which does not take many factors into account (immigration and emigration, for example) that can influence human populations to either grow or decline, nevertheless turned out to be fairly accurate in predicting the population. (iii)\)At \(t = 3,\,N = 20000\).Substituting these values into \((iii)\), we obtain\(20000 = {N_0}{e^{\frac{3}{2}(\ln 2)}}\)\({N_0} = \frac{{20000}}{{2\sqrt 2 }} \approx 7071\)Hence, \(7071\)people initially living in the country. Integrating with respect to x, we have y2 = 1 2 x2 + C or x2 2 +y2 = C. This is a family of ellipses with center at the origin and major axis on the x-axis.-4 -2 2 4 An ordinary differential equation (frequently called an "ODE," "diff eq," or "diffy Q") is an equality involving a function and its derivatives. I have a paper due over this, thanks for the ideas! They realize that reasoning abilities are just as crucial as analytical abilities.