applications of ordinary differential equations in daily life pdf

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Firstly, l say that I would like to thank you. which can be applied to many phenomena in science and engineering including the decay in radioactivity. One of the most basic examples of differential equations is the Malthusian Law of population growth dp/dt = rp shows how the population (p) changes with respect to time. %\f2E[ ^' in which differential equations dominate the study of many aspects of science and engineering. Differential equations have a variety of uses in daily life. Applications of Matrices and Partial Derivatives, S6 l04 analytical and numerical methods of structural analysis, Maths Investigatory Project Class 12 on Differentiation, Quantum algorithm for solving linear systems of equations, A Fixed Point Theorem Using Common Property (E. 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}$. By whitelisting SlideShare on your ad-blocker, you are supporting our community of content creators. Q.2. Here, we assume that $N(t)$is a differentiable, continuous function of time. If the object is large and well-insulated then it loses or gains heat slowly and the constant k is small. @ Packs for both Applications students and Analysis students. endstream endobj 87 0 obj <>stream Finding the series expansion of d u _ / du dk 'w\ Moreover, these equations are encountered in combined condition, convection and radiation problems. A differential equation is a mathematical statement containing one or more derivatives. Example Take Let us compute. Ordinary Differential Equations in Real World Situations Differential equations have a remarkable ability to predict the world around us. written as y0 = 2y x. One of the key features of differential equations is that they can account for the many factors that can influence the variable being studied. 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. A non-linear differential equation is defined by the non-linear polynomial equation, which consists of derivatives of several variables. 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. where the initial population, i.e. 2. Consider the dierential equation, a 0(x)y(n) +a If you are an IB teacher this could save you 200+ hours of preparation time. Bernoullis principle can be applied to various types of fluid flow, resulting in various forms of Bernoullis equation. You can then model what happens to the 2 species over time. The second order of differential equation represent derivatives involve and are equal to the number of energy storing elements and the differential equation is considered as ordinary, We learnt about the different types of Differential Equations and their applications above. hZ }y~HI@ p/Z8)wE PY{4u'C#J758SM%M!)P :%ej*uj-) (7Hh$Uh28~(4 Ordinary Differential Equations with Applications Authors: Carmen Chicone 0; Carmen Chicone. We thus take into account the most straightforward differential equations model available to control a particular species population dynamics. The applications of second-order differential equations are as follows: Thesecond-order differential equationis given by, \({y^{\prime \prime }} + p(x){y^\prime } + q(x)y = f(x)$. When students can use their math skills to solve issues they could see again in a scientific or engineering course, they are more likely to acquire the material. 40K Students Enrolled. (iii)\)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. 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. 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. Radioactive decay is a random process, but the overall rate of decay for a large number of atoms is predictable. Adding ingredients to a recipe.e.g. Change), You are commenting using your Twitter account. If you enjoyed this post, you might also like: Langtons Ant Order out ofChaos How computer simulations can be used to model life. What is the average distance between 2 points in arectangle? To solve a math equation, you need to decide what operation to perform on each side of the equation. Systems of the electric circuit consisted of an inductor, and a resistor attached in series, A circuit containing an inductance L or a capacitor C and resistor R with current and voltage variables given by the differential equation of the same form. An example application: Falling bodies2 3. Recording the population growth rate is necessary since populations are growing worldwide daily. A Super Exploration Guide with 168 pages of essential advice from a current IB examiner to ensure you get great marks on your coursework. The principal quantities used to describe the motion of an object are position ( s ), velocity ( v ), and acceleration ( a ). So l would like to study simple real problems solved by ODEs. (i)\)At $t = 0,\,N = {N_0}$Hence, it follows from $(i)$that $N = c{e^{k0}}$$ \Rightarrow {N_0} = c{e^{k0}}$$\therefore \,{N_0} = c$Thus, $N = {N_0}{e^{kt}}\,(ii)$At $t = 2,\,N = 2{N_0}$[After two years the population has doubled]Substituting these values into $(ii)$,We have $2{N_0} = {N_0}{e^{kt}}$from which $k = \frac{1}{2}\ln 2$Substituting these values into $(i)$gives\(N = {N_0}{e^{\frac{t}{2}(\ln 2)}}\,. These show the direction a massless fluid element will travel in at any point in time. 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. It thus encourages and amplifies the transfer of knowledge between scientists with different backgrounds and from different disciplines who study, solve or apply the . 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. 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. Surprisingly, they are even present in large numbers in the human body. We regularly post articles on the topic to assist students and adults struggling with their day to day lives due to these learning disabilities. hZqZ$[ |Yl+N"5w2*QRZ#MJ 5Yd`3V D;) r#a@ An ordinary differential equation (also abbreviated as ODE), in Mathematics, is an equation which consists of one or more functions of one independent variable along with their derivatives. We've encountered a problem, please try again. Get some practice of the same on our free Testbook App. endstream endobj 83 0 obj <>/Metadata 21 0 R/PageLayout/OneColumn/Pages 80 0 R/StructTreeRoot 41 0 R/Type/Catalog>> endobj 84 0 obj <>/ExtGState<>/Font<>/XObject<>>>/Rotate 0/StructParents 0/Type/Page>> endobj 85 0 obj <>stream Thus when it suits our purposes, we shall use the normal forms to represent general rst- and second-order ordinary differential equations. [11] Initial conditions for the Caputo derivatives are expressed in terms of Ive also made 17 full investigation questions which are also excellent starting points for explorations. 2.2 Application to Mixing problems: These problems arise in many settings, such as when combining solutions in a chemistry lab . Differential equations can be used to describe the rate of decay of radioactive isotopes. 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. Wikipedia references: Streamlines, streaklines, and pathlines; Stream function <quote> Streamlines are a family of curves that are instantaneously tangent to the velocity vector of the flow. Atoms are held together by chemical bonds to form compounds and molecules. equations are called, as will be defined later, a system of two second-order ordinary differential equations. Ordinary dierential equations frequently occur as mathematical models in many branches of science, engineering and economy. Bernoullis principle can be derived from the principle of conservation of energy. The constant k is called the rate constant or growth constant, and has units of inverse time (number per second). %%EOF systems that change in time according to some fixed rule. 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). A differential equation is an equation that relates one or more functions and their derivatives. Some of the most common and practical uses are discussed below. THE NATURAL GROWTH EQUATION The natural growth equation is the differential equation dy dt = ky where k is a constant. So, here it goes: All around us, changes happen. In the field of medical science to study the growth or spread of certain diseases in the human body. Thefirst-order differential equationis given by. 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. which is a linear equation in the variable $y^{1-n}$. Phase Spaces3 . mM-65_/4.i;bTh#"op}^q/ttKivSW^K8'7|c8J Due in part to growing interest in dynamical systems and a general desire to enhance mathematics learning and instruction, the teaching and learning of differential equations are moving in new directions. 300 IB Maths Exploration ideas, video tutorials and Exploration Guides, February 28, 2014 in Real life maths | Tags: differential equations, predator prey. It has only the first-order derivative$\frac{{dy}}{{dx}}$. 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. 2Y9} ~EN]+E- }=>S8Smdr\_U[K-z=+m`{ioZ If so, how would you characterize the motion? CBSE Class 9 Result: The Central Board of Secondary Education (CBSE) Class 9 result is a crucial milestone for students as it marks the end of their primary education and the beginning of their secondary education. Linearity and the superposition principle9 1. More precisely, suppose j;n2 N, Eis a Euclidean space, and FW dom.F/ R nC 1copies E E! P3 investigation questions and fully typed mark scheme. If k < 0, then the variable y decreases over time, approaching zero asymptotically. 3.1 Application of Ordinary Differential Equations to the Model for Forecasting Corruption In the current search and arrest of a large number of corrupt officials involved in the crime, ordinary differential equations can be used for mathematical modeling To . This is useful for predicting the behavior of radioactive isotopes and understanding their role in various applications, such as medicine and power generation. Various strategies that have proved to be effective are as follows: Technology can be used in various ways, depending on institutional restrictions, available resources, and instructor preferences, such as a teacher-led demonstration tool, a lab activity carried out outside of class time, or an integrated component of regular class sessions. The most common use of differential equations in science is to model dynamical systems, i.e. Find amount of salt in the tank at any time $t$.Ans:Here, ${V_0} = 100,\,a = 20,\,b = 0$, and $e = f = 5$,Now, from equation $\frac{{dQ}}{{dt}} + f\left( {\frac{Q}{{\left( {{V_0} + et ft} \right)}}} \right) = be$, we get$\frac{{dQ}}{{dt}} + \left( {\frac{1}{{20}}} \right)Q = 0$The solution of this linear equation is $Q = c{e^{\frac{{ t}}{{20}}}}\,(i)$At $t = 0$we are given that $Q = a = 20$Substituting these values into $(i)$, we find that $c = 20$so that $(i)$can be rewritten as$Q = 20{e^{\frac{{ t}}{{20}}}}$Note that as $t \to \infty ,\,Q \to 0$as it should since only freshwater is added. Two dimensional heat flow equation which is steady state becomes the two dimensional Laplaces equation, $\frac{{{\partial ^2}u}}{{\partial {x^2}}} + \frac{{{\partial ^2}u}}{{\partial {y^2}}} = 0$, 4. Differential equations can be used to describe the relationship between velocity and acceleration, as well as other physical quantities. When a pendulum is displaced sideways from its equilibrium position, there is a restoring force due to gravity that causes it to accelerate back to its equilibrium position. Learn more about Logarithmic Functions here. Mathematics has grown increasingly lengthy hands in every core aspect. They are used to calculate the movement of an item like a pendulum, movement of electricity and represent thermodynamics concepts. The task for the lecturer is to create a link between abstract mathematical ideas and real-world applications of the theory. Q.2. They are represented using second order differential equations. So, our solution . Weve updated our privacy policy so that we are compliant with changing global privacy regulations and to provide you with insight into the limited ways in which we use your data. Under Newtons law of cooling, we can Predict how long it takes for a hot object to cool down at a certain temperature. Clipping is a handy way to collect important slides you want to go back to later. A good example of an electrical actuator is a fuel injector, which is found in internal combustion engines. Students must translate an issue from a real-world situation into a mathematical model, solve that model, and then apply the solutions to the original problem. It is often difficult to operate with power series. By solving this differential equation, we can determine the velocity of an object as a function of time, given its acceleration. Flipped Learning: Overview | Examples | Pros & Cons. 2022 (CBSE Board Toppers 2022): Applications of Differential Equations: A differential equation, also abbreviated as D.E., is an equation for the unknown functions of one or more variables. Where $k$is a positive constant of proportionality. Students are asked to create the equation or the models heuristics rather than being given the model or algorithm and instructed to enter numbers into the equation to discover the solution. Example 14.2 (Maxwell's equations). Electrical systems also can be described using differential equations. Graphical representations of the development of diseases are another common way to use differential equations in medical uses. Anscombes Quartet the importance ofgraphs! Discover the world's. In the description of various exponential growths and decays. An equation that involves independent variables, dependent variables and their differentials is called a differential equation. Similarly, we can use differential equations to describe the relationship between velocity and acceleration. The use of technology, which requires that ideas and approaches be approached graphically, numerically, analytically, and descriptively, modeling, and student feedback is a springboard for considering new techniques for helping students understand the fundamental concepts and approaches in differential equations. Thank you. 82 0 obj <> endobj }9#J{2Qr4#]!L_Jf*K04Je$~Br|yyQG>CX/.OM1cDk$~Z3XswC\pz~m]7y})oVM\\/Wz]dYxq5?B[?C J|P2y]bv.0Z7 sZO3)i_z*f>8 SJJlEZla>`4B||jC?szMyavz5rL S)Z|t)+y T3"M`!2NGK aiQKd` n6>L cx*-cb_7% H|TN#I}cD~Av{fG0 %aGU@yju|k.n>}m;aR5^zab%"8rt"BP Z0zUb9m%|AQ@ $47\(F5Isr4QNb1mW;K%H@ 8Qr/iVh*CjMa`"w By solving this differential equation, we can determine the number of atoms of the isotope remaining at any time t, given the initial number of atoms and the decay constant. Homogeneous Differential Equations are used in medicine, economics, aerospace, automobile as well as in the chemical industry. This restoring force causes an oscillatory motion in the pendulum. 2) In engineering for describing the movement of electricity According to course-ending polls, students undergo a metamorphosis once they perceive that the lectures and evaluations are focused on issues they could face in the real world. Such kind of equations arise in the mathematical modeling of various physical phenomena, such as heat conduction in materials with mem-ory. There are various other applications of differential equations in the field of engineering(determining the equation of a falling object. An ODE of order is an equation of the form (1) where is a function of , is the first derivative with respect to , and is the th derivative with respect to . Let T(t) be the temperature of a body and let T(t) denote the constant temperature of the surrounding medium. They realize that reasoning abilities are just as crucial as analytical abilities. 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 M for mass, P for population, T for temperature, and so forth. Hence, the order is $2$. Q.4. Since, by definition, x = x 6 . Find the equation of the curve for which the Cartesian subtangent varies as the reciprocal of the square of the abscissa.Ans:Let $P(x,\,y)$be any point on the curve, according to the questionSubtangent $ \propto \frac{1}{{{x^2}}}$or $y\frac{{dx}}{{dy}} = \frac{k}{{{x^2}}}$Where $k$ is constant of proportionality or $\frac{{kdy}}{y} = {x^2}dx$Integrating, we get $k\ln y = \frac{{{x^3}}}{3} + \ln c$Or $\ln \frac{{{y^k}}}{c} = \frac{{{x^3}}}{3}$${y^k} = {c^{\frac{{{x^3}}}{3}}}$which is the required equation. 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. where k is called the growth constant or the decay constant, as appropriate. 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. Examples of applications of Linear differential equations to physics. Begin by multiplying by y^{-n} and (1-n) to obtain, $(1-n)y^{-n}y+(1-n)P(x)y^{1-n}=(1-n)Q(x)$, ${d\over{dx}}[y^{1-n}]+(1-n)P(x)y^{1-n}=(1-n)Q(x)$. 208 0 obj <> endobj Now lets briefly learn some of the major applications. 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. )CO!Nk&$(e'k-~@gB`. This equation represents Newtons law of cooling. Thus ${dT\over{t}}$ > 0 and the constant k must be negative is the product of two negatives and it is positive. 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. This allows you to change the parameters (such as predator birth rate, predator aggression and predator dependance on its prey). Electric circuits are used to supply electricity. Covalent, polar covalent, and ionic connections are all types of chemical bonding. At $t = 0$, fresh water is poured into the tank at the rate of ${\rm{5 lit}}{\rm{./min}}$, while the well stirred mixture leaves the tank at the same rate. We assume the body is cooling, then the temperature of the body is decreasing and losing heat energy to the surrounding. It is fairly easy to see that if k > 0, we have grown, and if k <0, we have decay. 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. What is Dyscalculia aka Number Dyslexia? The above graph shows almost-periodic behaviour in the moose population with a largely stable wolf population. ) 4) In economics to find optimum investment strategies ) 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. In mathematical terms, if P(t) denotes the total population at time t, then this assumption can be expressed as. 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 Important topics including first and second order linear equations, initial value problems and qualitative theory are presented in separate chapters. Differential equations have aided the development of several fields of study. What is a differential equation and its application?Ans:An equation that has independent variables, dependent variables and their differentials is called a differential equation. Application of Partial Derivative in Engineering: In image processing edge detection algorithm is used which uses partial derivatives to improve edge detection. You can download the paper by clicking the button above. A Differential Equation and its Solutions5 . I have a paper due over this, thanks for the ideas! Examples of Evolutionary Processes2 . If you read the wiki page on Gompertz functions [http://en.wikipedia.org/wiki/Gompertz_function] this might be a good starting point. It appears that you have an ad-blocker running. $\frac{{{\partial ^2}T}}{{\partial {t^2}}} = {c^2}\frac{{{\partial ^2}y}}{{\partial {x^2}}}$, $\frac{{\partial u}}{{\partial t}} = {c^2}\frac{{{\partial ^2}T}}{{\partial {x^2}}}$, 3. 115 0 obj <>stream As is often said, nothing in excess is inherently desirable, and the same is true with bacteria. Several problems in engineering give rise to partial differential equations like wave equations and the one-dimensional heat flow equation. Summarized below are some crucial and common applications of the differential equation from real-life. By using our site, you agree to our collection of information through the use of cookies. Where, $k$is the constant of proportionality. Activate your 30 day free trialto unlock unlimited reading. (LogOut/ The equation that involves independent variables, dependent variables and their derivatives is called a differential equation. This Course. Ordinary Differential Equations are used to calculate the movement or flow of electricity, motion of an object to and fro like a pendulum, to explain thermodynamics concepts. They are used in a wide variety of disciplines, from biology, economics, physics, chemistry and engineering. I was thinking of using related rates as my ia topic but Im not sure how to apply related rates into physics or medicine. If the object is small and poorly insulated then it loses or gains heat more quickly and the constant k is large. As with the Navier-Stokes equations, we think of the gradient, divergence, and curl as taking partial derivatives in space (and not time t). The applications of partial differential equations are as follows: A Partial differential equation (or PDE) relates the partial derivatives of an unknown multivariable function. 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 A second-order differential equation involves two derivatives of the equation. A differential equation is an equation that contains a function with one or more derivatives. Chemical bonds are forces that hold atoms together to make compounds or molecules. All content on this site has been written by Andrew Chambers (MSc. Separating the variables, we get 2yy0 = x or 2ydy= xdx. Ordinary differential equations are applied in real life for a variety of reasons. Differential equations have a remarkable ability to predict the world around us. Malthus used this law to predict how a species would grow over time. 5) In physics to describe the motion of waves, pendulums or chaotic systems. In the field of engineering, differential equations are commonly used to design and analyze systems such as electrical circuits, mechanical systems, and control systems. To learn more, view ourPrivacy Policy. Ordinary Differential Equations An ordinary differential equation (or ODE) is an equation involving derivatives of an unknown quantity with respect to a single variable. 4) In economics to find optimum investment strategies 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. " BDi$#Ab`S+X Hqg h 6 Thefirst-order differential equationis defined by an equation$\frac{{dy}}{{dx}} = f(x,\,y)$, here $x$and $y$are independent and dependent variables respectively. Tap here to review the details. MODELING OF SECOND ORDER DIFFERENTIAL EQUATION And Applications of Second Order Differential Equations:- 2. 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. Finally, the general solution of the Bernoulli equation is, $y^{1-n}e^{\int(1-n)p(x)ax}=\int(1-n)Q(x)e^{\int(1-n)p(x)ax}dx+C$. \h@7v"0Bgq1z)/yfW,aX)iB0Q(M\leb5nm@I 5;;7Q"m/@o%!=QA65cCtnsaKCyX>4+1J`LEu,49,@'T 9/60Wm Example: ${d^y\over{dx^2}}+10{dy\over{dx}}+9y=0$Applications of Nonhomogeneous Differential Equations, The second-order nonhomogeneous differential equation to predict the amplitudes of the vibrating mass in the situation of near-resonant. They are used in a wide variety of disciplines, from biology The results are usually CBSE Class 7 Result: The Central Board of Secondary Education (CBSE) is responsible for regulating the exams for Classes 6 to 9. In general, differential equations are a powerful tool for describing and analyzing the behavior of physical systems that change over time, and they are widely used in a variety of fields, including physics, engineering, and economics. ]JGaGiXp0zg6AYS}k@0h,(hB12PaT#Er#+3TOa9%(R*%= ( xRg -a*[0s&QM Written in a clear, logical and concise manner, this comprehensive resource allows students to quickly understand the key principles, techniques and applications of ordinary differential equations. 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. In PM Spaces. This graph above shows what happens when you reach an equilibrium point in this simulation the predators are much less aggressive and it leads to both populations have stable populations. 231 0 obj <>stream This means that. Functions 6 5. 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). To browse Academia.edu and the wider internet faster and more securely, please take a few seconds toupgrade your browser. Click here to review the details. %PDF-1.5 % See Figure 1 for sample graphs of y = e kt in these two cases. This is the differential equation for simple harmonic motion with n2=km. Population Models Application Of First Order Differential Equation, Application Of Second Order Differential Equation, Common Applications of Differential Equations in Physics, Exponential Reduction or Radioactivity Decay, Applications of Differential Equations in Real Life, Application of Differential Equations FAQs, Sum of squares of first n-natural numbers.