This document discusses various methods for solving first order differential equations, including:
1. Variable separable methods where the equation can be written as a function of x multiplied by a function of y.
2. Homogeneous equations where both sides are homogeneous functions of the same degree.
3. Exact equations where there exists an integrating factor.
4. Equations that can be transformed to an exact or separable form through substitution.
5. Linear equations that can be solved using an integrating factor that is a function of x.
Lesson 15: Exponential Growth and Decay (Section 021 slides)Matthew Leingang
Many problems in nature are expressible in terms of a certain differential equation that has a solution in terms of exponential functions. We look at the equation in general and some fun applications, including radioactivity, cooling, and interest.
Lesson 15: Exponential Growth and Decay (Section 021 slides)Matthew Leingang
Many problems in nature are expressible in terms of a certain differential equation that has a solution in terms of exponential functions. We look at the equation in general and some fun applications, including radioactivity, cooling, and interest.
Integration Made Easy!
The derivative of a function can be geometrically interpreted as the slope of the curve of the mathematical function f(x) plotted as a function of x. But its implications for the modeling of nature go far deeper than this simple geometric application might imply. After all, you can see yourself drawing finite triangles to discover slope, so why is the derivative so important? Its importance lies in the fact that many physical entities such as velocity, acceleration, force and so on are defined as instantaneous rates of change of some other quantity. The derivative can give you a precise intantaneous value for that rate of change and lead to precise modeling of the desired quantity.
Integration Made Easy!
The derivative of a function can be geometrically interpreted as the slope of the curve of the mathematical function f(x) plotted as a function of x. But its implications for the modeling of nature go far deeper than this simple geometric application might imply. After all, you can see yourself drawing finite triangles to discover slope, so why is the derivative so important? Its importance lies in the fact that many physical entities such as velocity, acceleration, force and so on are defined as instantaneous rates of change of some other quantity. The derivative can give you a precise intantaneous value for that rate of change and lead to precise modeling of the desired quantity.
A tutorial on the Frobenious Theorem, one of the most important results in differential geometry, with emphasis in its use in nonlinear control theory. All results are accompanied by proofs, but for a more thorough and detailed presentation refer to the book of A. Isidori.
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry.
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
Francesca Gottschalk - How can education support child empowerment.pptxEduSkills OECD
Francesca Gottschalk from the OECD’s Centre for Educational Research and Innovation presents at the Ask an Expert Webinar: How can education support child empowerment?
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
Embracing GenAI - A Strategic ImperativePeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
3. Linear Non-linear Integrating Factor Separable Homogeneous Exact Integrating Factor Transform to Exact Transform to separable
4. The first-order differential equation is called separable provided that f(x,y) can be written as the product of a function of x and a function of y. (1) Variables Separable
5. Suppose we can write the above equation as We then say we have “ separated ” the variables. By taking h ( y ) to the LHS, the equation becomes
7. Example 1 . Consider the DE Separating the variables, we get Integrating we get the solution as or an arbitrary constant.
8. Example 2. Consider the DE Separating the variables, we get Integrating we get the solution as or an arbitrary constant.
9. Homogeneous equations Definition A function f ( x , y ) is said to be homogeneous of degree n in x , y if for all t , x , y Examples is homogeneous of degree is homogeneous of degree 2. 0.
10. A first order DE is called homogeneous if are homogeneous functions of x and y of the same degree . This DE can be written in the form where is clearly homogeneous of degree 0.
11. The substitution y = z x converts the given equation into “variables separable” form and hence can be solved. (Note that z is also a (new) variable,) We illustrate by means of examples.
12. Example 3. Solve the DE Let y = z x. Hence we get That is
14. We express the LHS integral by partial fractions. We get or an arbitrary constant. Noting z = y/x, the solution is: c an arbitrary constant or c an arbitrary constant
15. Working Rule to solve a HDE: 1. Put the given equation in the form 2. Check M and N are Homogeneous function of the same degree.
16. 5. Put this value of dy/dx into (1) and solve the equation for z by separating the variables. 6. Replace z by y/x and simplify. 4. Differentiate y = z x to get
17. Example 4. Solve the DE Let y = z x. Hence we get or Separating the variables, we get Integrating we get where cosec z – cot z = c x and c an arbitrary constant.
18. We shall now see that some equations can be brought to homogeneous form by appropriate substitution. Non-homogeneous equations Example 5 Solve the DE That is
19. We shall now put x = u+h , y = v+k where h, k are constants ( to be chosen). Hence the given DE becomes We now choose h, k such that Hence
20. Hence the DE becomes which is homogeneous in u and v . Let v = z u. Hence we get
25. EXACT DIFFERENTIAL EQUATIONS A first order DE is called an exact DE if there exits a function f ( x , y ) such that Here df is the ‘total differential’ of f ( x , y ) and equals
26. Hence the given DE becomes df = 0 Integrating, we get the solution as f ( x , y ) = c , c an arbitrary constant Thus the solution curves of the given DE are the ‘level curves’ of the function f ( x , y ) . Example 8 The DE is exact as it is d ( xy ) = 0 Hence the solution is: x y = c
27. Example 7 The DE is exact as it is d ( x 2 + y 2 ) = 0 Hence the solution is: x 2 + y 2 = c Example 9 The DE is exact as it is Hence the solution is:
28. Test for exactness Suppose is exact. Hence there exists a function f ( x , y ) such that Hence Assuming all the 2 nd order mixed derivatives of f ( x , y ) are continuous, we get
30. We saw a necessary condition for exactness is We now show that the above condition is also sufficient for M dx + N dy = 0 to be exact. That is, if then there exists a function f ( x , y ) such that
31. Integrating partially w.r.t. x, we get where g ( y ) is a function of y alone We know that for this f ( x , y ), …… . (*) Differentiating (*) partially w.r.t. y, we get = N gives
32. or … (**) We now show that the R.H.S. of (**) is independent of x and thus g ( y ) (and so f ( x , y )) can be found by integrating (**) w.r.t. y . = 0 Q.E.D.
33. Note (1) The solution of the exact DE d f = 0 is f ( x , y ) = c . Note (2) When the given DE is exact, the solution f ( x , y ) = c is found as we did in the previous theorem. That is, we integrate M partially w.r.t. x to get The following examples will help you in understanding this. We now differentiate this partially w.r.t. y and equating to N , find g ( y ) and hence g ( y ).
34. Example 8 Test whether the following DE is exact. If exact, solve it. Here Hence exact. Now
35. Differentiating partially w.r.t. y , we get Hence Integrating, we get (Note that we have NOT put the arb constant ) Hence Thus the solution of the given d.e. is or c an arb const.
36. Example 9 Test whether the following DE is exact. If exact, solve it. Here Hence exact. Now
37. Differentiating partially w.r.t. y , we get Hence Integrating, we get (Note that we have NOT put the arb constant ) Hence Thus the solution of the given d.e. is or c an arb const.
38. In the above problems, we found f ( x , y ) by integrating M partially w.r.t. x and then We can reverse the roles of x and y . That is we can find f ( x , y ) by integrating N partially The following problem illustrates this. w.r.t. y and then equate equated
39. Example 10 Test whether the following DE is exact. If exact, solve it. Here Hence exact. Now
40. Differentiating partially w.r.t. x , we get gives Integrating, we get (Note that we have NOT put the arb constant ) Hence Thus the solution of the given d.e. is or c an arb const.
41. The DE is NOT exact but becomes exact when multiplied by i.e. We say as it becomes is an Integrating Factor of the given DE Integrating Factors
42. Definition If on multiplying by ( x , y ), the DE becomes an exact DE, we say that ( x , y ) is an Integrating Factor of the above DE are all integrating factors of the non-exact DE We give some methods of finding integrating factors of an non-exact DE
43. Problem Under what conditions will the DE have an integrating factor that is a function of x alone ? Solution. Suppose = h ( x ) is an I.F. Multiplying by h ( x ) the above d.e. becomes Since (*) is an exact DE, we have
45. Hence if is a function of x alone, then is an integrating factor of the given DE
46. Rule 2: Consider the DE If , a function of y alone, then is an integrating factor of the given DE
47. Problem Under what conditions will the DE have an integrating factor that is a function of the product z = x y ? Solution. Suppose = h ( z ) is an I.F. Multiplying by h ( z ) the above d.e. becomes Since (*) is an exact DE, we have
49. Hence if is a function of z = x y alone, then is an integrating factor of the given DE
50. Example 11 Find an I.F. for the following DE and hence solve it. Here Hence the given DE is not exact.
51. Now a function of x alone. Hence is an integrating factor of the given DE Multiplying by x 2 , the given DE becomes
52. which is of the form Note that now Integrating, we easily see that the solution is c an arbitrary constant.
53. Example 12 Find an I.F. for the following DE and hence solve it. Here Hence the given DE is not exact.
54. Now a function of y alone. Hence is an integrating factor of the given DE Multiplying by sin y , the given DE becomes
55. which is of the form Note that now Integrating, we easily see that the solution is c an arbitrary constant.
56. Example 13 Find an I.F. for the following DE and hence solve it. Here Hence the given DE is not exact.
57. Now a function of z =x y alone. Hence is an integrating factor of the given DE
58. which is of the form Integrating, we easily see that the solution is c an arbitrary constant. Multiplying by the given DE becomes
59. Problem Under what conditions will the DE have an integrating factor that is a function of the sum z = x + y ? Solution. Suppose = h ( z ) is an I.F. Multiplying by h ( z ) the above DE becomes Since (*) is an exact DE, we have
61. Hence if is a function of z = x + y alone, then is an integrating factor of the given DE
62.
63. We assume that the function a 1 (x), a 0 (x), and b(x) are continuous on an interval and that a 1 (x) 0on that interval. Then, on dividing by a 1 (x), we can rewrite equation (1) in the standard form where P(x), Q(x) are continuous functions on the interval.
64. Let’s express equation (2) in the differential form If we test this equation for exactness, we find Consequently, equation(3) is exact only when P(x) = 0. It turns out that an integrating factor , which depends only on x, can easily obtained the general solution of (3).
65. Multiply (3) by a function (x) and try to determine (x) so that the resulting equation is exact. We see that (4) is exact if satisfies the DE Which is our desired IF
66. In (2), we multiply by (x) defined in (6) to obtain We know from (5) and so (7) can be written in the form
68. Working Rule to solve a LDE: 1. Write the equation in the standard form 2. Calculate the IF (x) by the formula 3. Multiply the equation in standard form by (x) and recalling that the LHS is just obtain
69. 4. Integrate the last equation and solve for y by dividing by (x).
70. Ex 1. Solve Solution :- Dividing by x cos x, throughout, we get
72. Problem (2g p. 62): Find the general solution of the equation Ans.:
73. The usual notation implies that x is independent variable & y the dependent variable. Sometimes it is helpful to replace x by y and y by x & work on the resulting equation. * When diff equation is of the form