Summary
One of the main goals of the University of Computer Sciences (UCI) is the training of skills that allow the independent preparation of students; establishing solid links between the development of science in today's society.
mathematical-modeling-differential-equationsAmong the many subjects taught throughout the career is included Mathematics III, whose essential objectives are to contribute to the creation of a solid base of knowledge, skills and values that will allow the student an adequate projection in their working life. as a Computer Engineer.
The main objective of this work is to encourage research ability in students by proposing an investigative work structure that contributes to the profile of the graduate. Emphasizing mathematical modeling and solving social problems.
Key words: Ordinary Differential Equations, mathematical modeling, problem solving.
Introduction
Under the heading University Graduate in Computer Sciences, more than 50 subjects are taught over the course of five years, in which to offer adequate training to the graduate so that he or she can enter the labor market in institutions and companies.
In recent years, thousands of jobs related to Computer Science have been created in our country in different companies and organizations. This gives an idea of the growing importance that computing is taking on in today's society. In fact, one cannot speak of a modern and up-to-date company if it has not incorporated adequate computerization of the process into its internal requirements. This demand continues to grow, to the point that it is estimated that in the next five years a large number of jobs related to this discipline will be created.
The University of Computer Sciences arose in the heat of the Battle of Ideas with the aim of meeting this demand that today's society demands; the computerization of the country is one of the fundamental objectives of the center. To achieve this task, it is necessary to train a quality university graduate who is capable of solving the different situations created in his job and in his daily life.
Among the many subjects taught throughout the career, Mathematics III, which includes among its topics Numerical Series, Series of Functions, Ordinary Differential Equations, Vector and Gradient Functions, Multiple Integrals and Line Integrals has essential objectives that contribute to the creation of a solid base of knowledge, skills and values that will allow the student an adequate projection in his working life as a Computer Engineer.
General Educational Objectives of Mathematics III.
As a Computer Engineer, the UCI graduate must be able to mathematically model real-life problems, and then be able to computerize them and deliver a final product.
In order to contribute to the formation of the graduate, Mathematics III draws up the following General Educational Objectives.
1. Contribute to the formation of a scientific conception of the world and scientific thought by understanding how a mathematical model is made and how it is a reflection of reality.
2. Contribute to the formation of the student's personality by developing habits of reflective thinking and critically evaluating the results of their work.
3. Contribute to the development of students' cognitive abilities, habits of using scientific literature, reasoning ability and thinking logically through the use of the subjects of the subject.
4. Contribute to the computational training of students through the introduction and use of scientific information techniques.
To meet these objectives, we must create values in students such as:
- Independence.
- Creativity.
- Personal growth.
- Projection into the future.
- Active position before the tasks assigned to it.
- Perseverance.
In addition to those ethical and aesthetic values that characterize our socialist society: Responsibility, revolutionary dignity, honesty, modesty, a critical spirit, solidarity, disinterest and patriotism.
Topic to develop,.
The present work was carried out in order to create an Active Investigative Practice that contributes to the objectives of the subject, for that we will initially focus on one of the themes that comprise it, Ordinary Differential Equations.
This topic consists of the following objectives:
1. To develop capacities to characterize and interpret the most important concepts and theorems related to Differential Equations.
2. Solve technical problems that are modeled through Ordinary Differential Equations (EDO).
What is currently happening?
The way of teaching the subject does not pay correctly to the objectives set, since the students are very dependent on the teacher, they do not have an investigative initiative, they do not worry about their personal improvement, they are unable to create a future projection. This implies that the fundamental objectives of the subject are not being achieved; And what is affecting the most is the lack of motivation on the part of the teacher, who encourages the student to do research, which is one of the most efficient activities to show the student how much she can do on her own.
Extra Class Homework Proposal
To achieve the objectives and values outlined above and thereby contribute positively to the training of a better graduate, in addition to seeking a complement that helps the teacher to achieve adequate motivation in the student towards the subject, so that he himself arrives at the conclusion of what it is for him to receive mathematics as part of his basic training; The idea arises to create a proposal for an Investigative Practical Activity.
This Activity will initially focus on the topic of Ordinary Differential Equations and consists of the following aspects.
The student must be able to:
- Investigate in which fields it is possible to apply the content given in classes.
- Identify when a problem from real life, taken entirely from society can be mathematically modeled through the use of Ordinary Differential Equations.
- Model the problem using the resources given in classes.
- Integrate the knowledge acquired both in the subjects of the discipline and in the rest of the subjects taught so far.
- Give a solution to the problem using the methods studied in class.
The task will be oriented in the first class of the course and will be discussed 1 week after the partial evaluation of the EDO topic.
The evaluation will be carried out by teams of at most 3 and at least 2 people.
The discussion will take place in 15 minutes using a Power Point Presentation of no more than 10 slides and showing mastery of the use of the mathematical assistants.
Structure of the ExtraClass Task.
An investigative work will be carried out consisting of a report whose format is described below:
Letter Arial 12, line 1.5, no more than 8 pages that do not include the annexes or the bibliography.
1. Presentation
2. Summary of 150 words, in Spanish and English.
3. Introduction.
- Existing problem (eg problem that exists in company or body.).
- What problem to solve (what part do you have to solve as a computer engineer).
- Theoretical foundations necessary to solve the problem (In addition to the EDOs, what other subject that you have received is applicable to the solution of the problem, in such a way that interdisciplinary integration is shown).
4. Development.
- Mathematical modeling of the selected problem with its scientific arguments.
- Resolution of the problem. A comparison should be made between a method learned in class and another method not received during the course. In addition, you must be able to use and associate the problem with the applications taught in classes or others.
- Results and analysis thereof (meaning of the value obtained, Interpretation).
5. Conclusions
- What was the activity for?
- What did you learn during your development?
- Found link of the subject with the Computer profile.
6. Bibliography
- References from the bibliography consulted, using the ISO 690 standard found at // ucistore / infotecnologia.
7. Annexes
Some areas of knowledge where the Ordinary Differential Equations are applicable.
- Growth and Decrease
The initial value problem
(1)
Where k is a constant of proportionality, it is used as a model of different phenomena where growth or decrease (disintegration) intervene.
In biology, it has been observed that in short periods the growth rate of some populations (such as those of bacteria or small animals) is proportional to the population present at any time. If we know a population at some arbitrary initial moment, which we can consider defined by, the solution of (1) helps us to predict the population in the future, this is for.
In physics, an initial value problem like equations (1) can serve as a model for roughly calculating the residual amount of a substance that decays or decays radioactively. That differential equation (1) can also describe the temperature of a cooling object.
In chemistry, the residual amount of a substance in certain reactions follows equation (1).
The constant of proportionality k, in (1), can be found by solving the initial value problem, with a determination of x in a moment.
- Radiocarbon Dating
Around 1950, chemist Willard Libby invented a method that uses radioactive carbon to determine approximate fossil ages.
The theory of radiocarbon dating (dating or dating) is based on the fact that the carbon 14 isotope is produced in the atmosphere by the action of cosmic radiation on nitrogen. The ratio of the amount of C-l4 to ordinary carbon in the atmosphere appears to be constant, and consequently the proportional amount of the isotope present in all living organisms is the same as that in the atmosphere. When an organism dies the absorption of C-l4 either by respiration or food ceases. Thus, by comparing the proportional amount of C-14 present, for example in a fossil, with the constant relationship that exists in the atmosphere, it is possible to obtain a reasonable estimate of its age. The method is based on the fact that the average period of radioactive C-l4 is known to be approximately 5600 years.
For this work, Libby won the Nobel Prize in chemistry in 1960. Her method was used to date the wooden furniture in Egyptian tombs and the linen wraps of the Dead Sea scrolls.
- Spring and Mass Systems: Free Motion Undamped
Hooke's Law: Suppose that, as in the figure, a mass is attached to a flexible spring hanging from a rigid support. When replaced with a different mass, the stretch, elongation, or elongation of the spring will change.
According to Hooke's law, the spring itself exerts a restoring force, opposite to the direction of elongation and proportional to the amount of elongation s. Specifically, where k is a constant of proportionality called the spring constant. Although masses with different weights stretch a spring in different amounts, it is essentially characterized by its number k; for example, if a mass weighing 10 pounds stretches a spring, then it implies that. So, necessarily, a mass weighing 8 pounds will stretch the foot spring.
Second law of Newton
After attaching a mass to a spring, it stretches it one length and reaches a balance position, in which its weight is balanced by the restoring force. Remember that weight is defined by, where the mass is expressed in slugs, kilograms or grams and, respectively.
As can be seen in the figure, the equilibrium condition is. If the mass is displaced a distance from its equilibrium position, the spring restoring force is Assuming that there are no retarding forces acting on the system and that the mass is moving free of other external forces (free movement), then We can equate Newton's second law with the net, or resultant, force of the restitution force and the weight:
(1)
The negative sign in equation (1) indicates that the spring restoring force acts in the opposite direction of movement. Furthermore, we can adopt the convention that the displacements measured below the equilibrium position are positive without stretching.
Differential equation of undamped free motion If we divide equation (1) by the mass m, we will obtain the second-order differential equation
(2)
where. Equation (2) is said to describe simple harmonic motion or undamped free motion. Two obvious initial conditions associated with (2) are, the amount of initial displacement, and, the initial speed of the mass. For example, yes, the mass starts from a point below the equilibrium position with an upward velocity. Yes, the mass is released starting from rest from a point located units above the equilibrium position, etc.
Solution and equation of motion
To solve equation (2), note that the solutions of the auxiliary equation are complex numbers. Thus the general solution of (2) is:
(3)
The period of free vibrations described by (3) is, and the frequency is. For example, for, the period is and the frequency is.
The previous number indicates that the graph of each unit is repeated and the last number indicates that there are three cycles of the graph each unit or, what is the same, that the mass goes through complete vibrations per unit of time. Furthermore, it can be shown that the period is the interval between two successive maxima of. Keep in mind that a maximum of is the positive displacement when the mass reaches the maximum distance below the equilibrium position, while a minimum of is the negative displacement when the mass reaches the maximum height above that position. Both cases are called extreme displacement of the mass. Finally, when the initial conditions are used to determine the constants in equation (3), the particular solution that results is said to be the equation of motion.
Problems included in the previously described areas of knowledge whose solutions lead to EDO.
In this section we present a sample of real problems with a pedagogical approach, which contribute to the fulfillment of the objectives of mathematical subject III. The student must be able to investigate in which area of knowledge she is, and in her turn identify which mathematical elements of those studied in classes will help her to model the problem to give her a mathematical solution that can be interpreted in the identified. area of knowledge.
problems
1. In a biotechnology laboratory, research is being done on a bacterial culture that is thought to be used in the development of a cancer vaccine. The research team among other properties of the culture needs to know the growth of the bacteria for whatever the initial amount of bacteria that the culture has. For this they have proposed to determine what is the necessary time that must elapse to triple the initial amount of microorganisms, taking into account that an hour has elapsed, that the measured amount of bacteria is and the reproduction ratio is proportional to the amount of bacteria present.
2. In China it is known that its population increases with a ratio proportional to the number of people it has at any time. In order to plan the feeding of the population, they carried out a study on the growth of the population and it turned out that in 5 years it doubled. Concerned planners wonder how long the population will triple and quadruple.
3. In a biology experimental center, a culture of bacteria is studied whose quantity of microorganisms at any given time grows at a rate proportional to the bacteria present. Biologists without knowing how many bacteria the culture had initially put it under observation and determined that after three hours there are 400 individuals. After 10 hours, there are 2000 specimens. To be able to dictate a specific result on the growth of bacteria, they will need to know what the initial amount of bacteria was when they started the grizzly.
4. On a work trip, a group of archaeologists found a fossilized bone that they think dates from the time of the Maya. In order to verify their assumptions the team dedicated all their efforts to determine the age of the fossil, during the analysis they discovered that it contained one hundredth of the original amount of C-14, this turned out to be a great advance pass for them, they already had all the data they needed, it only remained to determine the age of the fossil.
5. When a vertical ray of light passes through a transparent substance, the rate at which its intensity decreases is proportional to where t represents the thickness, in feet, of the medium. In clear sea water, the intensity below the surface is 25% of the initial intensity of the incident ray. What is the intensity of the beam below the surface?
6. When interest is continuously compounded (or compounded), at any time the amount of money increases at a rate proportional to the present amount: where is the annual interest rate.
a) Calculate the amount collected at the end of five years, when $ 5000 is deposited in a savings account that yields continuously compounded annual interest.
b) In how many years will the initial capital have doubled?
7. In some cases, when two parallel constant springs hold a single counterweight, the effective spring constant of the system is
A counterweight stretches one spring and another. These springs are fixed to a common rigid support at their upper part and to a metal plate at their lower end. As seen in the figure, the counterweight is fixed to the center of the system board. Determine the effective spring constant of this system. Deduce the equation of motion, if the counterweight starts from the equilibrium position, with a downward velocity.
Bibliography
- ADDINE, FÁTIMA (et al) (1998) Didactics and optimization of the teaching-learning process. __: Havana: IPLAC (In electronic support) BIXIO, CECILIA (1999) Teaching to learn. Rosario: Homo Sapienshttp: //teleformacion.uci.cu/course/view.php? Id = 36DENNIS G. ZILL_ Loyola Marymount University: sixth edition (1997) Differential Equations with Modeling ApplicationsMedellin Milan, Pedro and Caraveo, Luz María. "Curricular development in the construction of an academic project", Higher Education Journal, No. 91, July-September, pp. 51-60. 1994 Mertens, Leonardo "Management by labor competence in the Company and vocational training". Organization of Ibero-American States, for science and culture. Digital library. Year 2002. Author Granville, "Differential and integral calculus", editorial LIMUSA,ISBN 968-18-1178-XAutores Berkley / Blanchard, «Calculos», Saunders College PublishingGómez, P. (2002). Didactic analysis and curricular design in mathematics. EMA Magazine, 7 (3), 251-292, Flores, P. (2001). Class as a context for academic tasks (working document). Quinn, M. (2002). Qualitative research and evaluation methods. Thousand Oaks, CA: Sage Publications, Inc.
