![International Journal of Humanities and Social Science Invention
ISSN (Online): 2319 – 7722, ISSN (Print): 2319 – 7714
www.ijhssi.org ||Volume 6 Issue 1||January. 2017 || PP.24-27
www.ijhssi.org 24 | Page
An Experimental Setup for Teaching Newton's Law of Cooling
J. Genaro Gonzalez-Hernandez1
, Cesar Medellin-Verduzco2
1,2
Department of Mechatronics, Technological University of Altamira, Mexico
ABSTRACT: When differential equations are presented in most of Mexican universities mathematics courses,
it is common that students have some problems identifying relationship between these equations and real world.
In this work an experimental setup for teaching the most important features of Newton's law of cooling is
presented. A simple electronic circuit is implemented in order to let the students contrast the solution of the
linear differential equation that characterizes the phenomenon with results obtained in circuit measurements. In
order to validate the effectiveness of the proposed method, experimental and control groups are set.
Keywords: Teaching, learning, cooling dynamics, academic performance, electric circuit
I. INTRODUCTION
An age of great technological advances has leaded the modern world, especially in electronics,
computer sciences and robotics, disciplines that have been used effectively not only to improve the quality of
life worldwide, but also as a teaching methodology in schools [1]-[4]. The advances in science and technology
can also be used to transmit efficiently the knowledge to students through appropriate practical and computer-
based activities which allow them to corroborate theories exposed in daily life [5].
In Mexican universities, the field of mathematics is frequently handled without considering appropriate
practices to prove equations an concepts in the real world, although there are many interesting phenomena
where mathematics can be applied. Among differential equations exist several important applications, one of
them is related with Newton's law of cooling, which is presented in many common phenomena, for example in
the manufacture of an industrial metallic piece and in the cooling of a cake fresh from the oven [6].
Precedents which caused this work have their origin in difficulty presented when teaching differential
equations in university programs, because of dryness presented for lack practical implementation with elements
that students can detect in a tangible and explicit way. This problem has been reported by several university
teachers who have taught the subject of differential equations during several generations. All this generated the
idea to use electronic components to create a design that allows a computer to obtain temperature readings and
built graphs which can be compared against the theoretical curve.
The general objective was to improve comprehension of students related to Newton's law of cooling
by verifying results obtained theoretically and those obtained in real measurements. To accomplish the goal, a
didactic prototype using electronic components for verifying temperature variation of a body exposed to an
environment with a different temperature was made. It was also important to couple the signal of ADC to PC to
storage data and for its subsequent interpretation [7]. It was essential to select the experimental and control
groups in order to compare academic performance in both at the end of the test.
The use of a didactic prototype that allows the verification of Newton's law of cooling facilitates
students comprehension in this subject of differential equations, so it saves time in data acquisition and graphs
generation, showing an automatic verification system between practical and theoretical cooling curves. These
developments justify the implementation of physical, electronic and computational elements that students can
perform to motivate their own knowledge, generating a propitious environment to raise the level of
understanding of this important subject in real world and make the class more pleasant.
Nowadays teaching of mathematics is not stock-still, but it has changed to more dynamic shapes that
involve students participation, so they can obtain their knowledge by reasoning which is generated through
interesting and even funny activities applied in daily life. Pedagogy should allow to build and acquire
knowledge moreover lead mathematical thinking and logical, numerical and space reasoning, like so awake a
natural interest while facing challenges for the solution of problems [8].
Learning mathematics at any level implies an attractive challenge at all times, and assuming it ideally
should be enjoyable and not be a heavy burden for the student, but rather be taken as an opportunity where the
search of solutions pursues no other objective than to give satisfaction to those around it [9].
Metacognitive processes, which invite individual reflection and have a unifying character that underlies
a profound conception of the learning processes, cannot be achieved without the optimal conditions to trigger
the natural flow of intellectual processes in the individual. In this sense, the efforts of teachers should be aimed
at creating the appropriate learning environments to arouse the student's interest in the programmatic contents of](https://image.slidesharecdn.com/e0601042427-170228105528/85/An-Experimental-Setup-for-Teaching-Newton-s-Law-of-Cooling-1-320.jpg)
![An Experimental Setup for Teaching Newton's Law of Cooling
www.ijhssi.org 25 | Page
the subjects, and one of the ways to achieve it is through the experimental implementation that corroborates
theories exposed in class [10].
The theory-based school interventions are very different from those based on practice, the theorist
whose work is summarized in the text only tries to break down the basic laws of memory, which would only
allow explaining the mechanical repetition of meaningless concepts. Even in these days, echoes of these
situations are found in a great quantity of materials designed to improve the mental abilities; there is also an
important number of teachers who resort to their own knowledge and who rely on popularizing methods that
will improve mental processes [11]. The inclination that has been adopted in questions of thought, underlies the
theory of multiple intelligences, which has been adopted in the last decades [12].
Most math teachers generally show a marked tendency towards the use of annoying and arid
algorithms, instead of establishing practical, simple and interesting examples, they present their students real
enigmas whose meaning is not understood or deciphered however, the contrast between theoretical mathematics
and its relation to practice constitutes a very high step in the understanding of concepts and awakens interest in
the majority of students [13].
Many people have the deep-seated idea that mathematics is boring, cold and difficult, although some
have a very different view of them as a warm, beautiful and exciting subject, it is all a matter of focus and the
fact that mathematical relationships are in the natural world and that men live immersed in them, constitutes an
undeniable truth that will be used by the mathematician to offer those who have a negative concept of this
important science, a portal that allows to change their perspective by revealing their presence everywhere, thus
corroborating its application in practice [14].
This concept in which mathematics is presented as a living, tangible and regulating entity of the
relationships that exist between the various elements that constitute the life of man, becomes evident through the
17 equations that changed the world. This set of equations revel a special brightness that should emerge in the
students by meditating on the importance of numerical relationships and their characterization in the
mathematical models that explain the reality of the world in which they live [15].
II. METHODOLOGY
During the course of the differential equations, two groups of students were selected, one that received
the classes in a traditional way, and another which was treated using the proposed teaching-learning method.
Each group was of 20 students, which were examined at the end of the exercise to measure their understanding
of Newton's law of cooling.
The students were taken from groups already established, but with very similar general characteristics,
since they were students of professional level and of the same degree, in this way, a possible bias generated by
its heterogeneity diminished. The first thing that was done, was to explain the problem of Newton's law of
cooling in the classroom, then the differential equation was modeled and solved. In addition some exercises
were performed. All this was applied for the two work groups.
Afterwards, the students of the experimental group had a practical experience in which they verified
the behavior of the cooling system of Newton. Working groups were formed of four students, who implemented
a circuit that was constructed and managed with the following materials: a resistance of 47 Ω to 5 W, an
experimental board, a switch, a temperature sensor LM35, a multimeter, a power source, a data acquisition
board, a PC and electrical wire as it is shown in Fig. 1.
The thermal radiation of the resistance due to the current flow falls on the body of the LM35, which
sends a voltage signal proportional to the temperature (10 mV for each degree centigrade), which is transformed
by the ADC to a digital format and it is also sent to the computer for its processing.
Once the circuit was built, the students first measured the temperature of the environment and recorded
it (TA), then closed the switch and waited until the resistor temperature was 75°C, once this was accomplished,
the switch was opened, so the initial temperature T(0) was 75°C. Five minutes later the new body temperature
was measured, then with the initial data T(0), the ambient temperature TA, and the temperature recorded at five
minutes T(5), the students calculated the constants of the exponential equation that solves the differential
equation, (1), where "t" represents the time, "T(t)" is the resistor temperature in function of time, "c" and "k" are
constants of the system, and "TA" is the environment temperature [16].
A
kt
TcetT )(](https://image.slidesharecdn.com/e0601042427-170228105528/85/An-Experimental-Setup-for-Teaching-Newton-s-Law-of-Cooling-2-320.jpg)
![An Experimental Setup for Teaching Newton's Law of Cooling
www.ijhssi.org 27 | Page
Fig. 2. Comparison between graphics of recorded and calculated temperature.
IV. CONCLUSION
The students were able to interpret that the results presented an important relationship between the
graph of the temperature recorded and the theoretical solution curve of the differential equation, they also
noticed that the discrepancies between them are in the most critical case of less than two degrees Centigrade,
and that were generated due to the thermal radiation itself that affects the environment, altering the temperature
thereof.
The ease of automatically collating the graphs allowed students to quickly observe that the rate of
change in temperature of a body is directly proportional to the difference between that temperature and that of
the environment. The results of the obtained evaluation indicate a greater academic performance in the students
who verified the problem of cooling in the practice, especially in relation to the understanding of the dynamics
of the system.
It is important to mention that these conclusions are based on the numerical values obtained from the
evaluation instruments applied and shown in Table 2, but they not received rigorous statistical treatment to
ensure that there were significant differences due to the application of the method. It is considering to be
implemented in future researches.
REFERENCES
[1] F. Shahnia, M. Moghbel, and H. H. Yengejeh, Motivating Power System Protection Course Students Practical and Computer-Based
Activities, IEEE Transactions on Education, 59(2), 2016, 81-90.
[2] J.J. Olarte, C. Dominguez, A. Jaime and F. J. Garcia-Izquierdo, Students and Staff Perceptions of Key As[ects of Computer Science
Engineering Capstone Projects, IEEE Transactions on Education, 59(1), 2016, 45-51.
[3] A. Ruiz-Iniesta, G. Jimenez-Diaz, A. Jaime and M. Gomez-Albarran, A Semantically Enriched Contex-Aware OER Recomendation
Strategy and Its Aplication to a Computer Science OER Repository, IEEE Transactions on Education, 57(4), 2016, 255-260.
[4] J. McLurkin, J. Rykowski, M. John, Q. Kaseman, and A. J. Lynch, Using Multi-Robot Systems for Engineering Education:
Teaching and Outreach With Large Numbers of an Advanced, Low-Cost Robot, IEEE Transactions on Education, 56(1), 2013, 24-
33.
[5] S. Nikolic, P.J. Vial, M. Ros, D. Stirling, and C. Ritz, Improving the Laboratory Learning Experience: A Process to Train and
Manage Teaching Assistants, IEEE Transactions on Education, 58(2), 2015, 130-139.
[6] M. Golubitsky, and M. Dellnitz, Linear Algebra and Differential Equations Using MATLAB (Mexico, DF: International Thomson,
2001).
[7] V. Giri, A Pedagogically Effective Structured Introduction to Electrical Energy Systems with Coupled Laboratory Experiences,
IEEE Transactions on Power Systems, 19(1), 2004, 129-138.
[8] A. Cofre, and L. Tapia, Recreational Mathematics in the Classroom: Proposes to Make Classes More Pleasant (Mexico, DF:
Alfaomega, 2006).
[9] M.C. Chamorro, Didactic in Mathematics (Madrid, Spain: Pearson, 2005).
[10] R. Cantoral, O. Covian, R.M. Farfan, J. Lezama, and A. Romo, Researches Concerning to Teaching and Learning of Mathematics
(Mexico, DF: Diaz de Santos, 2008).
[11] H. Gardner, Multiple Intelligences. Theory in Practice (Barcelona, Spain: Paidos Iberica, 2005).
[12] D. Gamon, and A. Bragdon, Inteligent Excercises. Conditioning Excercices for the Six Types of Intelligence (Mexico, DF: Tomo,
2008).
[13] M. Tahan, Curious and Funny Mathematics (Mexico, DF: Oceano, 2009).
[14] J.A. De-la-Peña, Algebra Everywhere (Mexico, DF: Fondo de Cultura Economica, 2004).
[15] I. Stewart, In Pursuit of the Unknown (New York, NY: Basic Books, 2012).
[16] D. Zill, Differential Equations with Applications (Mexico, DF: Iberoamericana, 1997).
[17] C. Dolores, G. Martinez, R.M. Farfan, C. Carrillo, I. Lopez, and C. Navarro, Educative Mathematics (Mexico, DF: Diaz de Santos,
2007).](https://image.slidesharecdn.com/e0601042427-170228105528/85/An-Experimental-Setup-for-Teaching-Newton-s-Law-of-Cooling-4-320.jpg)
An Experimental Setup for Teaching Newton's Law of Cooling
- 1. International Journal of Humanities and Social Science Invention ISSN (Online): 2319 – 7722, ISSN (Print): 2319 – 7714 www.ijhssi.org ||Volume 6 Issue 1||January. 2017 || PP.24-27 www.ijhssi.org 24 | Page An Experimental Setup for Teaching Newton's Law of Cooling J. Genaro Gonzalez-Hernandez1 , Cesar Medellin-Verduzco2 1,2 Department of Mechatronics, Technological University of Altamira, Mexico ABSTRACT: When differential equations are presented in most of Mexican universities mathematics courses, it is common that students have some problems identifying relationship between these equations and real world. In this work an experimental setup for teaching the most important features of Newton's law of cooling is presented. A simple electronic circuit is implemented in order to let the students contrast the solution of the linear differential equation that characterizes the phenomenon with results obtained in circuit measurements. In order to validate the effectiveness of the proposed method, experimental and control groups are set. Keywords: Teaching, learning, cooling dynamics, academic performance, electric circuit I. INTRODUCTION An age of great technological advances has leaded the modern world, especially in electronics, computer sciences and robotics, disciplines that have been used effectively not only to improve the quality of life worldwide, but also as a teaching methodology in schools [1]-[4]. The advances in science and technology can also be used to transmit efficiently the knowledge to students through appropriate practical and computer- based activities which allow them to corroborate theories exposed in daily life [5]. In Mexican universities, the field of mathematics is frequently handled without considering appropriate practices to prove equations an concepts in the real world, although there are many interesting phenomena where mathematics can be applied. Among differential equations exist several important applications, one of them is related with Newton's law of cooling, which is presented in many common phenomena, for example in the manufacture of an industrial metallic piece and in the cooling of a cake fresh from the oven [6]. Precedents which caused this work have their origin in difficulty presented when teaching differential equations in university programs, because of dryness presented for lack practical implementation with elements that students can detect in a tangible and explicit way. This problem has been reported by several university teachers who have taught the subject of differential equations during several generations. All this generated the idea to use electronic components to create a design that allows a computer to obtain temperature readings and built graphs which can be compared against the theoretical curve. The general objective was to improve comprehension of students related to Newton's law of cooling by verifying results obtained theoretically and those obtained in real measurements. To accomplish the goal, a didactic prototype using electronic components for verifying temperature variation of a body exposed to an environment with a different temperature was made. It was also important to couple the signal of ADC to PC to storage data and for its subsequent interpretation [7]. It was essential to select the experimental and control groups in order to compare academic performance in both at the end of the test. The use of a didactic prototype that allows the verification of Newton's law of cooling facilitates students comprehension in this subject of differential equations, so it saves time in data acquisition and graphs generation, showing an automatic verification system between practical and theoretical cooling curves. These developments justify the implementation of physical, electronic and computational elements that students can perform to motivate their own knowledge, generating a propitious environment to raise the level of understanding of this important subject in real world and make the class more pleasant. Nowadays teaching of mathematics is not stock-still, but it has changed to more dynamic shapes that involve students participation, so they can obtain their knowledge by reasoning which is generated through interesting and even funny activities applied in daily life. Pedagogy should allow to build and acquire knowledge moreover lead mathematical thinking and logical, numerical and space reasoning, like so awake a natural interest while facing challenges for the solution of problems [8]. Learning mathematics at any level implies an attractive challenge at all times, and assuming it ideally should be enjoyable and not be a heavy burden for the student, but rather be taken as an opportunity where the search of solutions pursues no other objective than to give satisfaction to those around it [9]. Metacognitive processes, which invite individual reflection and have a unifying character that underlies a profound conception of the learning processes, cannot be achieved without the optimal conditions to trigger the natural flow of intellectual processes in the individual. In this sense, the efforts of teachers should be aimed at creating the appropriate learning environments to arouse the student's interest in the programmatic contents of
- 2. An Experimental Setup for Teaching Newton's Law of Cooling www.ijhssi.org 25 | Page the subjects, and one of the ways to achieve it is through the experimental implementation that corroborates theories exposed in class [10]. The theory-based school interventions are very different from those based on practice, the theorist whose work is summarized in the text only tries to break down the basic laws of memory, which would only allow explaining the mechanical repetition of meaningless concepts. Even in these days, echoes of these situations are found in a great quantity of materials designed to improve the mental abilities; there is also an important number of teachers who resort to their own knowledge and who rely on popularizing methods that will improve mental processes [11]. The inclination that has been adopted in questions of thought, underlies the theory of multiple intelligences, which has been adopted in the last decades [12]. Most math teachers generally show a marked tendency towards the use of annoying and arid algorithms, instead of establishing practical, simple and interesting examples, they present their students real enigmas whose meaning is not understood or deciphered however, the contrast between theoretical mathematics and its relation to practice constitutes a very high step in the understanding of concepts and awakens interest in the majority of students [13]. Many people have the deep-seated idea that mathematics is boring, cold and difficult, although some have a very different view of them as a warm, beautiful and exciting subject, it is all a matter of focus and the fact that mathematical relationships are in the natural world and that men live immersed in them, constitutes an undeniable truth that will be used by the mathematician to offer those who have a negative concept of this important science, a portal that allows to change their perspective by revealing their presence everywhere, thus corroborating its application in practice [14]. This concept in which mathematics is presented as a living, tangible and regulating entity of the relationships that exist between the various elements that constitute the life of man, becomes evident through the 17 equations that changed the world. This set of equations revel a special brightness that should emerge in the students by meditating on the importance of numerical relationships and their characterization in the mathematical models that explain the reality of the world in which they live [15]. II. METHODOLOGY During the course of the differential equations, two groups of students were selected, one that received the classes in a traditional way, and another which was treated using the proposed teaching-learning method. Each group was of 20 students, which were examined at the end of the exercise to measure their understanding of Newton's law of cooling. The students were taken from groups already established, but with very similar general characteristics, since they were students of professional level and of the same degree, in this way, a possible bias generated by its heterogeneity diminished. The first thing that was done, was to explain the problem of Newton's law of cooling in the classroom, then the differential equation was modeled and solved. In addition some exercises were performed. All this was applied for the two work groups. Afterwards, the students of the experimental group had a practical experience in which they verified the behavior of the cooling system of Newton. Working groups were formed of four students, who implemented a circuit that was constructed and managed with the following materials: a resistance of 47 Ω to 5 W, an experimental board, a switch, a temperature sensor LM35, a multimeter, a power source, a data acquisition board, a PC and electrical wire as it is shown in Fig. 1. The thermal radiation of the resistance due to the current flow falls on the body of the LM35, which sends a voltage signal proportional to the temperature (10 mV for each degree centigrade), which is transformed by the ADC to a digital format and it is also sent to the computer for its processing. Once the circuit was built, the students first measured the temperature of the environment and recorded it (TA), then closed the switch and waited until the resistor temperature was 75°C, once this was accomplished, the switch was opened, so the initial temperature T(0) was 75°C. Five minutes later the new body temperature was measured, then with the initial data T(0), the ambient temperature TA, and the temperature recorded at five minutes T(5), the students calculated the constants of the exponential equation that solves the differential equation, (1), where "t" represents the time, "T(t)" is the resistor temperature in function of time, "c" and "k" are constants of the system, and "TA" is the environment temperature [16]. A kt TcetT )(
- 3. An Experimental Setup for Teaching Newton's Law of Cooling www.ijhssi.org 26 | Page Fig. 1. Experimental setup to prove differential equation related to Newton's law of cooling. Once the constants were obtained, the students calculated the temperature values that the resistor would have at 10, 15, 20, 25 and 30 minutes. Then the temperatures were recorded every 5 minutes and the results of the measurements were compared with those predicted by the mathematical model. Finally, after a few days an evaluation was applied both to the students of the control group and to the experimentation group, which consisted of three sections, the first one had to do with the understanding of the dynamics of the system, the second referred to problems where it was necessary to calculate temperatures and the last one in relation to problems where the calculation of the times was required. III. FINDINGS AND COMMENTS Table 1 presents the temperature values recorded and calculated, as well as the deviations thereof, taking as reference an ambient temperature of 30°C. Fig. 2 shows a comparison between graphics of recorded and calculate temperature. Table 1. Registered and calculated temperatures in the experiment. Time (minutes) Recorded temperature (°C) Calculated temperature (°C) Variation (°C) 0 75 75 0 5 48.8 48.8 0 10 38.4 37.85 0.55 15 34.6 33.28 1.32 20 33.1 31.37 1.73 25 32.5 30.57 1.93 30 32.2 30.23 1.97 As regards the students' performance of the experimental and control groups in relation to Newton's cooling problem, Table 2 presents the results, where the understanding column of the dynamics of the system refers to the first section of the test applied to both groups, whose questions were directed to the understanding of the mathematics of variation (Dolores et al., 2007) and the behavior of the phenomenon itself. The temperature calculation column shows the results of the second part of the test, where in the problems the time was given and the body temperature was requested. Finally, the time calculation column corresponds to the third part of the evaluation, where the data is the body temperature and it is necessary to calculate the time at which the temperature is reached. Each section consisted of five questions, while the total duration of the evaluation was one hour. Table 2. Averages of tests applied per indicator in each group. Group Indicator of comprehension and application of concepts System dynamics comprehension Temperature calculation Time calculation A (traditional method) 73 68 62 B (proposed method) 93 82 75
- 4. An Experimental Setup for Teaching Newton's Law of Cooling www.ijhssi.org 27 | Page Fig. 2. Comparison between graphics of recorded and calculated temperature. IV. CONCLUSION The students were able to interpret that the results presented an important relationship between the graph of the temperature recorded and the theoretical solution curve of the differential equation, they also noticed that the discrepancies between them are in the most critical case of less than two degrees Centigrade, and that were generated due to the thermal radiation itself that affects the environment, altering the temperature thereof. The ease of automatically collating the graphs allowed students to quickly observe that the rate of change in temperature of a body is directly proportional to the difference between that temperature and that of the environment. The results of the obtained evaluation indicate a greater academic performance in the students who verified the problem of cooling in the practice, especially in relation to the understanding of the dynamics of the system. It is important to mention that these conclusions are based on the numerical values obtained from the evaluation instruments applied and shown in Table 2, but they not received rigorous statistical treatment to ensure that there were significant differences due to the application of the method. It is considering to be implemented in future researches. REFERENCES [1] F. Shahnia, M. Moghbel, and H. H. Yengejeh, Motivating Power System Protection Course Students Practical and Computer-Based Activities, IEEE Transactions on Education, 59(2), 2016, 81-90. [2] J.J. Olarte, C. Dominguez, A. Jaime and F. J. Garcia-Izquierdo, Students and Staff Perceptions of Key As[ects of Computer Science Engineering Capstone Projects, IEEE Transactions on Education, 59(1), 2016, 45-51. [3] A. Ruiz-Iniesta, G. Jimenez-Diaz, A. Jaime and M. Gomez-Albarran, A Semantically Enriched Contex-Aware OER Recomendation Strategy and Its Aplication to a Computer Science OER Repository, IEEE Transactions on Education, 57(4), 2016, 255-260. [4] J. McLurkin, J. Rykowski, M. John, Q. Kaseman, and A. J. Lynch, Using Multi-Robot Systems for Engineering Education: Teaching and Outreach With Large Numbers of an Advanced, Low-Cost Robot, IEEE Transactions on Education, 56(1), 2013, 24- 33. [5] S. Nikolic, P.J. Vial, M. Ros, D. Stirling, and C. Ritz, Improving the Laboratory Learning Experience: A Process to Train and Manage Teaching Assistants, IEEE Transactions on Education, 58(2), 2015, 130-139. [6] M. Golubitsky, and M. Dellnitz, Linear Algebra and Differential Equations Using MATLAB (Mexico, DF: International Thomson, 2001). [7] V. Giri, A Pedagogically Effective Structured Introduction to Electrical Energy Systems with Coupled Laboratory Experiences, IEEE Transactions on Power Systems, 19(1), 2004, 129-138. [8] A. Cofre, and L. Tapia, Recreational Mathematics in the Classroom: Proposes to Make Classes More Pleasant (Mexico, DF: Alfaomega, 2006). [9] M.C. Chamorro, Didactic in Mathematics (Madrid, Spain: Pearson, 2005). [10] R. Cantoral, O. Covian, R.M. Farfan, J. Lezama, and A. Romo, Researches Concerning to Teaching and Learning of Mathematics (Mexico, DF: Diaz de Santos, 2008). [11] H. Gardner, Multiple Intelligences. Theory in Practice (Barcelona, Spain: Paidos Iberica, 2005). [12] D. Gamon, and A. Bragdon, Inteligent Excercises. Conditioning Excercices for the Six Types of Intelligence (Mexico, DF: Tomo, 2008). [13] M. Tahan, Curious and Funny Mathematics (Mexico, DF: Oceano, 2009). [14] J.A. De-la-Peña, Algebra Everywhere (Mexico, DF: Fondo de Cultura Economica, 2004). [15] I. Stewart, In Pursuit of the Unknown (New York, NY: Basic Books, 2012). [16] D. Zill, Differential Equations with Applications (Mexico, DF: Iberoamericana, 1997). [17] C. Dolores, G. Martinez, R.M. Farfan, C. Carrillo, I. Lopez, and C. Navarro, Educative Mathematics (Mexico, DF: Diaz de Santos, 2007).