Geometrical Application Of Ordinary Differential Equation

Geometrical Application Of Ordinary Differential Equation Many practical problems in science and engineering are formulated by finding how one quantity is related to, or depends upon, one or more (other) quantities defined In the problem. Often, it is easier to model a relation between the rates of changes in the variable rather than between the variables themselves. This study of this relationship gives rise to differential equation. Derivatives can always be interpreted as rate. For example, if x is a function of t then dx/dt is the rate of x with respect to t. if x denotes the displacement of a particle, then dx/dt represents the velocity of the particle. If x represents the electric charge then dx/dt represents the flow of charge that is the current. Derivatives of higher orders represents rate of rates. If x denotes the displacement of particle, then d2x/dt2 represents the accelerations. A differential equation can be defined as an equation containing derivatives of various orders and variables .differential equation which involves one independent variable are called ordinary differential equation. If the differential equation involves more than one independent variable and partial derivatives of the dependent variable with respect to them, than it is called partial differential equation. Explanation:- Let y be the dependent variable and x be the independent variable. So the system can be denoted as dy/dx= y , d2y/dx2=y Some Example of Ordinary Differential equation y=62 y+16y =2x x2y-xy+6y=log x yy+ y2 = x2 Introduction to differential equation, and solving linear differential equations using operator method:- In this Term paper, I will first introduce what differential equation is? Separable first order differential equation will be solved. Then the integrating factor will be taught to solve linear differential equation of the first degree. The auxiliary equation (or characteristic equation) will be introduced to solve homogeneous linear equations, and then operator method will be taught finally to solve non-homogeneous linear equations. This term paper assumes readers familiar with basic of calculus, like differentiation and integration. What is differential equation? A differential equation is an equation which contains derivatives. Here are some examples: In these equations, y is an unknown function depends on x which we would like to solve. These kind of equations are very important in different fields, like in chemistry describing rate of reaction, physics describing equation of motion, etc. Therefore, able to solve these equations analytically enables us to understand many natural process. The above equations are known as ordinary differential equations(ODE) since they only contain derivatives with respect to one variable, x. (note that the equations hold for all values of x) In mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable. A simple example is Newtons second law of motion, which leads to the differential equation for the motion of a particle of constant mass m. In general, the force F depends upon the position x(t) of the particle at time t, and thus the unknown function x(t) appears on both sides of the differential equation, as is indicated in the notation F(x(t)). Ordinary differential equations are distinguished from partial differential equations, which involve partial derivatives of functions of several variables. Ordinary differential equations arise in many different contexts including geometry, mechanics, astronomy and population modelling. Many famous mathematicians have studied differential equations and contributed to the field, including Newton, Leibniz, the Bernoulli family, Riccati, Clairaut, dAlembert and Euler. Much study has been devoted to the solution of ordinary differential equations. In the case where the equation is linear, it can be solved by analytical methods. Unfortunately, most of the interesting differential equations are non-linear and, with a few exceptions, cannot be solved exactly. Approximate solutions are arrived at using computer approximations. The trajectory of a projectile launched from a cannon follows a curve determined by an ordinary differential equation that is derived from Newtons second law. Ordinary differential equation Let y be an unknown function in x with y(n) the nth derivative of y, and let F be a given function then an equation of the form is called an ordinary differential equation (ODE) of order n. If y is an unknown vector valued function , it is called a system of ordinary differential equations of dimension m (in this case, F : à ¢Ã¢â‚¬Å¾Ã‚ mn+1à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ à ¢Ã¢â‚¬Å¾Ã‚ m). More generally, an implicit ordinary differential equation of order n has the form where F : à ¢Ã¢â‚¬Å¾Ã‚ n+2à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ à ¢Ã¢â‚¬Å¾Ã‚  depends on y(n). To distinguish the above case from this one, an equation of the form is called an explicit differential equation. A differential equation not depending on x is called autonomous. A differential equation is said to be linear if F can be written as a linear combination of the derivatives of y together with a constant term, all possibly depending on x: with ai(x) and r(x) continuous functions in x. The function r(x) is called the source term; if r(x)=0 then the linear differential equation is called homogeneous, otherwise it is called non-homogeneous or inhomogeneous. Solutions Given a differential equation a function u: I à ¢Ã…  Ã¢â‚¬Å¡ R à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ R is called the solution or integral curve for F, if u is n-times differentiable on I, and Given two solutions u: J à ¢Ã…  Ã¢â‚¬Å¡ R à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ R and v: I à ¢Ã…  Ã¢â‚¬Å¡ R à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ R, u is called an extension of v if I à ¢Ã…  Ã¢â‚¬Å¡ J and A solution which has no extension is called a global solution. Terminology Partial differential equations These are equations which involves more than one independent variable. For instance: Partial differential equations(PDE) are significantly more difficult than ODE, and we wont talk about it at this moment. Order Order of a differential equations is the order of the highest derivative in the equation. Order 1: Order 2: Degree The degree of a differential equation is the degree of the highest derivative in the equation. Degree 1: Degree 2: Separable 1st order ODE If the ODE is in the following form, the solution can be found using integration easily: Example:- In the study of partial differentiation, recall that a function of two variables that equals a constant, describes the points in the 3-D plane with the same potential; . The curves that connect the points with the same potential are called level curves and have the value of c. A contour map is a level curve graph where common elevations are connected giving a 2-D representation of a 3-D reality. Using a LiveMath 3-D graph theory you can plot such a function along with the level curves describing the contours associated with that function. The picture below uses the following function to demonstrate this (here is a LiveMath plug-in animation of the graph below). In general terms, this type of equation is represented by the following: It describes the level curves and is the solution to the following differential equation. The equation below is just the total derivative of the function above. Because it is the total derivative of some function z(x,y) it is called an Exact Differential Equation. To help understand how to solve these types of equations you will look at the solution first and then analyze how to back into that solution. In this example you will take the total derivative of a function and analyze its parts. Then you will take this new equation (a differential equation now) and, knowing the answer, describe the method used to solve it. Input the following equation: To take a total derivative in LiveMath, first input the differential operator d times z (d*z). Input this into a second Prop and substitute the equation into it. Collect common terms on the RHS and Expand the coefficients of the differentials for the final answer. After setting the RHS equal to zero you will have a differential equation to solve. Notice how the coefficients are neither separable, homogeneous, nor are they linear. To help analyze this equation, label the coefficient of dx as M and the coefficient of dy as N. Both are functions of x and y so place the equation in the following form. The total derivative of a function is obtained by adding the partial derivatives of the coefficients. This is done with the equation below. Set up our notebook in the following manner: Perform the substitutions to give the partial derivatives. we can see that differential equations of this type are Exact. They are immediate derivatives of another function. You know this is true in this example because you developed the equation below by taking the derivative of the original function. we can describe an Exact Differential Equation as an equation whose dx coefficient is the partial derivative with respect to x of some function f(x,y) and whose dy coefficient is the partial derivative with respect to y of the SAME function. Since you dry-labed the last example you know what this equation, z=f(x,y), is. This will not be the case as you look to solve these problems though, so you need to find a way of determining that an equation is exact, then you will know that M and N are related to the solution equation in this way! Using the fact that these partial derivatives are of the same function will be the key to the method used to solve these equations. To test a differential equation for exactness, follow the method described in the next example. Test for Exactness This example demonstrates the test for exactness of the same equation used in the previous example. First input the differential equation as shown below. Remember to include an Independence Declaration inside the same case theory the test is performed. To test for exactness, equate the partial derivative with respect to y of M and the partial derivative with respect to x of N. Notice that these partials are with respect to the exact opposite variables as those used to determine the total derivative in the last example. The reason for this will become clear to you later when you derive this test. Set up the partial derivatives and solve by substituting M and N into the partial derivative Ops. The fact that they are equal means that the differential equation is exact! Method of Solution: To solve these types of equations you will need to take one or the other coefficient and go backwards to determine the solution. You can take either M or N to do this, it is up to you. For this example try using M. Solution Method First, set up an equation equating the unknown function, named , to an integral of M plus some unknown function of y. Call this function u for the time being. The reason you do this is the fact that to get M, the partial derivative was taken of the unknown function with respect to x. You will try to back into the answer by integrating M. This is not automatic though, because of the fact that when a partial derivative is performed, one of the variables is treated as a constant and therefore drops out (the derivative of a constant is = 0). Below this derivative is displayed again. We will not get back the function by integrating M, because the y term is not there! It is the constant, as is shown below where you try to get the function back by integrating M. This is very close to the answer, and with a little twist, will lead to a method that you will use to obtain the solution. Input the following props and perform the substitutions as shown. The user defined variable u is used in this case, rather than the arbitrary constant c, because you are actually looking for, what you might call, an arbitrary function. It will also be necessary later to have u defined as a variable for LiveMath to solve for the function. Now we have a potential function (), that represents the solution to the problem. To solve, the function u must be determined. If you take the partial derivative of the unknown function with respect to y this time, you will get N. By setting up the equation this way, you can then isolate u. You already know what N is, so: Next substitute the potential function into the Prop and solve for u by performing an integration. The final solution is achieved by substituting this u Prop back into the function . Since this function describes level curves, it is set equal to a constant c. The question remains, why do you take partial derivatives of M and N to determine if an equation is exact? M and N have been defined as the partial derivatives of z with respect to x and y respectively. By taking the second partial derivative of each coefficient WITH RESPECT TO THE OPPOSITE VARIABLE, the LHS of both of these equations is equal and therefore the RHS are equal too.

Officer Selection

Because of the range of duties, officers should possess certain traits: hectically agility, the ability to cope with difficult situations, well-developed writing skills, good communication skills, sound Judgment, compassion, strong powers of observation, and the ability to both exert and respect commands of authority. Minimum Requirements Every department sets its own standards when considering candidates for police officers, however most departments require a series of minimum standards which perspective applicants must have.All applicants must be at least 21 years of age and have or be eligible to receive a driver's license because their primary duty Is patrol, ND they must be able to drive to respond to Incidents. Police officers must also be able to possess a firearm. In order to qualify to own a firearm, a person must be at least twenty-one years old. Applicants must also have no Felony convictions. Convicted felons also are prohibited from possessing a firearm, which thereby ba rs them from becoming police officers.Individuals with domestic violence convictions are no longer able to possess a firearm, thereby prohibiting them from becoming police officers as well (Grant & Terry, 2009). Finally many police departments now have educational standards for recruits. Nearly all departments require officers to have at least a high school diploma. And many require at least some college credits. Written Examination The written examination is the first step in becoming a police officer once a formal application has been submitted.The test varies by department, but It might be a civil service exam, an exam produced by the individual police department, or one produced by a private testing company. The exam does not test specific legal or criminal Justice knowledge, but rather evaluates the candidate's basic reading, writing, and comprehension skills. The exam will likely contain a number of different sections, whereby the candidate must be able to understand and write In English, write a sample essay, understand basic mathematics, memorize facts, show sound 1 OFF reasoning Ana logic, Ana analyze potential scenarios .For clamatorial ten written examination is developed by the California Commission on Peace Officer Standards and Training (POST) and measures reading comprehension and writing abilities (caperers. Com, 2011). Departmental Interview Departmental Interview will evaluate the applicant's interpersonal skills, problem solving, oral communication and other abilities not tested by other examination components. The interview is not scored; however, the interview panel will make recommendations regarding who should proceed in the final hiring process.The interview can be structured, unstructured, or a combination or both. In a structured interview, the candidate is asked a series of questions regarding the Job and his or her specific abilities. Structured questions such as â€Å"Do you drink alcohol†, usually require specified answers direct answers. The alternative to this would be to conduct a semi structured interview with open ended questions on particular topics. Structured interviews allow for a better comparison of candidates on specific topics, pen-ended questions are likely to elicit more information.Though the candidate must pass all phases of the selection process in order to be hired as a police officer, the interview process is critical in the assessment of the candidate's attitudes, appearance, and demeanor. Physical Ability Examination The Physical Ability Examination will measure physical performance through a series of exercises that will be administered on a pass/fail basis. Measuring a police candidate's level of physical agility is a crucial part of the selection process, although the physical agility test has been controversial and has undergone significant hangs since its inception (Grant & Terry, 2009).The Physical Ability Examination has gone through many changes. Until the sass's the test required applicants to demonstrate substantial upper body strength which kept many women from completing the test successfully thereby eliminating them from the candidate pool. The introduction of Title VII in 1972 as well as the Equal Employment Opportunity Commission (EEOC) guidelines on sex discrimination barred the refusal to hire a female applicant because of characteristics attributed to women as a class and thus the physical agility tests have changed considerably in the past few decades.Psychological and Polygraph Tests A Polygraph Examination is used to verify the veracity and accuracy of information submitted by candidates regarding, but not necessarily limited to: use of controlled substances; driving, criminal, medical and employment history; and other Job-related factors. The polygraph works by recording involuntary physiological changes in the body that occur when a person is partaking in conscious deceit.The purpose of the psychological screening process is to measur e intelligence and to identify personality characteristics and any mental disorders that may lead to problematic behavior in he future (Grant & Terry, 2009). Psychological screening, particularly those measuring conscientiousness, emotional stability, agreeableness, and integrity – have been shown to aid in the prediction of on-the-Job performance across a wide variety of occupations, including peace officers (POST. A. Gob, 2011) It is important to screen out individuals who may exhibit mental or personality deficits, because police officers interact with individuals on a daily basis and often in high-stress situations. Background Investigation I en employment, connecter Ana Docudrama Investigation consists AT a tongue duty of the candidate's history prior to appointment to determine fitness for this employment.Reasons for rejection include use of controlled substances, felony convictions, repeated or serious violations of the law, inability to work cooperatively with co-work ers, inability to accept supervision, or other relevant factors. Candidates who are disqualified during the background investigation process must wait two years from the date of disqualification before they may reapply to take the Police Officer examination.Candidates who are disqualified because of uncorrectable deiced problems, serious drug abuse or because of criminal records may possibly not be allowed to reapply. Training Once a police candidate has passed through the selection process, he or she is hired on probation, a trial period of one or two years during which the officer is evaluated. This probationary period begins with training at the police academy, a school where officers learn on-the-Job techniques prior to receiving full police powers.Officers must train at the academy for up to 1,100 hours, and they receive full pay and benefits from the time they enter the academy (Grant & Terry, 2009). Training is rigorous, demanding and exhausting. It is also a rewarding life-c hanging experience. New officers learn how much they are capable of by succeeding at seemingly impossible challenges, both physical and mental Mainland. Com, 2011). While in the academy, the officer receives educational as well as practical physical The Los Angles Police Department (LAPS) Academy Curriculum includes training.Academics, which encompasses arrest and booking procedures, preliminary investigation techniques, radio and communications, report writing, traffic investigation, and traffic enforcement, Driving, which includes emergency procedures ND defensive driving techniques, Firearms Training, which trains candidates in effective and safe use of police issued firearms, Law, which covers search and seizure, evidence, laws of arrest, crimes against persons and property, sex crimes, crimes against children, and other general criminal statutes falling under the California Penal Code, Los Angles Municipal Code, Welfare and Institutions Code, and Federal Laws, and finally physi cal training which builds strength and endurance through physical conditioning while promoting a positive attitude toward a fitness lifestyle. It also encompasses training in physical arrest techniques, controls, and weaponless defense Mainland. Com, 2011).Development Once a new police officer leaves the academy, they are assigned a field training officer (FOOT) who assists the new officer to acclimate into the police culture, or experience the solicitation process. Solicitation involves learning the values, social processes, and behaviors associated with the police institution. It involves the patterns of interaction that depend on the relations of individuals in particular settings (Grant & Terry, 2009). Foot can have a significant influence over new officers ND assist them in dealing with the inevitable stress and cynicism of the Job. Conclusion Selecting qualified police officers is a lengthy, competitive process, involving multiple phases. Candidates are exposed too battery of tests both physically and mentally to ascertain their overall qualifications and abilities.

7 aportaciones de los hispanos a Estados Unidos

A Fragile Civilization in The Lord of the Flies by William...

The lord of the flies is a novel by William Golding author published in 1954 that shows fragility of civilization. It describes the regressive course of children themselves. After a plane crash, a group of children found alone without adults on a deserted island. Quickly the group organized in a democratic pattern: they choose by-election a leader, Ralph, and decide the role of each. Meetings organized, privileged moments lyrics. Various incidents and life which looks tougher as they thought initially will gradually switch the group into savagery and tyranny, symbolized by another character lighthouse, Jack. The wild pig hunt reveals youth from the primitive impulses. And the fire, show their presence from the sea, goes one day because†¦show more content†¦He is big, strong and dark and wants to impose by force. contradictory, he brings the food to the group by hunting. He awakens in others and he urges of violence. And he manipulates others of terror, by establishing ritual and beliefs. It symbolizes the brutality, tyranny. Piggy, the intellectual to the ungrateful physical and makes asthma heckled for his weakness but respected for intelligence and usefulness. His glasses that symbolize knowledge have enabled the group to have fire, essential for survival. He is the guarantor of the values of the civilization. He knows think, analyses the situation is Seer. It symbolizes wisdom and civilization. Simon, the martyr, seeking truth is one that symbolizes wisdom and will be the first victim of the madness that will gradually win the group to him. It was he who discovers the truth about the monster of the island (it is a dead paratrooper). Wanting to warn others of their mistake and gets killed (know the truth to go into the unknown). Ralph is the one who brings together all children from the island by the calling with the conch (shell). He became Chief and establishes rules of life. It symbolizes the civilisation and the democracy. From the perspective of a political vision, this novel shows the failure of civilization and democracy. This island novel is an ideal laboratory for the analysis of the human species. It is a testing ground for new forms of life in society, by destruction of the pre-established frameworksShow MoreRelatedLord Of The Flies : Representation Of Violence And War1611 Words   |  7 PagesLord Of The Flies: Representation Of Violence and War Dietrich Bonhoeffer, a German theologian, states that â€Å" The ultimate test of a moral society is the kind of world that it leaves to its children.† In William Golding’s Lord Of The Flies, societal topics run rampant throughout the text with Golding’s use of individuals to represent different aspects of society. Many writers view the Lord Of The Flies as an allegory, as societal topics such as politics make appearances throughout the text. InRead MoreMichelle Duan Mrs. MJ English 10 H, per. 3 13 February 2014 A Symbol’s Worth a Thousand1500 Words   |  6 Pagesis the nature of the symbols found in William Golding’s Lord of the Flies. As a group of boys stranded on an island struggle to survive without adult supervision to maintain order, Golding uses a variety of objects to convey their descent from civilization into brutality, violence, and savagery. Of these objects, three hold particular significance. In Lord of the Flies, Golding uses the conch, the signal fire, and the Lord of the Flies to symbolize civilization, hope for rescue, and inner evil whileRead MoreSymbolism In Teh Kite Runner And Lord Of The Flies1102 Words   |  5 Pagesrunner and lord of the flies. Khaled Hosseini, in Teh Kite Runner, and William Golding, in Lord of the Flies, Bothe use symbols in their narratives to represent broader ideas like politics, Hope and Evil, but the interpretation pf these presented ideas is varied. Both Khaled Hosseini and William Golding use a symbol to represent politics in their respective narratives. William Golding uses the Island as a parallel to Britain and how it is viewed from different individuals during WWII. Golding expressesRead MoreThe Symbolism of the Conch Shell in Lord of the Flies by William Golding1086 Words   |  5 PagesThe Symbolism of the Conch In Lord of the Flies, several symbols are used to illustrate important ideas that are crucial to the plot and meaning of the book. One of these symbols is the conch: this rare shell is not only a precious and expensive in the world of merchandise; it also holds a dark and mysterious power over a group of English boys, lost on an island with no adults, clues, or means of escape. The boys set up a civilization and try to live in the society they have set up. This systemRead More Essay on Behavior in All Quiet on the Western Front and Lord of the Flies1313 Words   |  6 PagesBehavior in All Quiet on the Western Front and Lord of the Flies  Ã‚         An authors view of human behavior is often reflected in their works. The novels All Quiet on the Western Front by Erich Maria Remarque and Lord of the Flies by William Golding are both examples of works that demonstrate their authors view of man, as well his opinion of war. Goldings Lord of the Flies is highly demonstrative of Goldings opinion that society is a thin and fragile veil that when removed shows man for whatRead MoreSocietal Breakdown On The Island1720 Words   |  7 PagesSocietal breakdown on the island in ‘Lord of the Flies’ is due to the inherent evil of man 3.8: Develop an informed understanding of literature and/or language using critical texts. Hypothesis: Societal breakdown on the island is due to the inherent evil of man Jason Carvalho ‘Lord of the Flies’ is the name of William Golding’s historically famous novel, yet it is more than just a title. It is a kind of statement, a way of mocking the very existences of humanity. Reading this book I cameRead MoreCan Lord of the Flies (William Golding) be Classified as a Fable?2254 Words   |  10 Pagesmoral instruction and its characters and scenes are drawn to suit this purpose. William Golding has referred to his novel, Lord of the Flies, as a fable. This essay will demonstrate that in the moral lessons it offers us and in the symbolic nature of its setting, characters and literary devices, the novel functions as a fable for the inherent tendency in man to revert to primal savagery once he is removed from civilization. We are left with the caution that evil must be acknowledged and consciouslyRead MoreSymbolism in Lord of the Flies by William Golding1153 Words   |  5 PagesGonzalo Barril Merino 3EMC Lord of the Flies Essay Describe the use of symbolism in Lord of the Flies By understanding symbols, you get a better picture of the novel â€Å"Lord of the Flies† and the hidden messages and references to human nature and a criticism of society. The author, William Golding, uses a huge amount of symbolism to reflect society of the outer world with the island. Symbols of fire, the conch and water are described all throughout the novel. Fire represents hope, strength and knowledgeRead MoreSymbolism in Lord of the Flies by William Golding1159 Words   |  5 PagesGonzalo Barril Merino 3EMC Lord of the Flies Essay Describe the use of symbolism in Lord of the Flies By understanding symbols, you get a better picture of the novel â€Å"Lord of the Flies† and the hidden messages and references to human nature and a criticism of society. The author, William Golding, uses a huge amount of symbolism to reflect society of the outer world with the island. Symbols of fire, the conch and water are described all throughout the novel. Fire represents hope, strengthRead MoreLord Of The Flies Literary Analysis1546 Words   |  7 Pagesoverruns them and evil starts to lurk over the island.The fictional story of the group of British schoolboys stranded on an island and the decisions they make, relates back to our society and the decisions we might make in a difficult situation. Lord of The Flies is Golding’s attempt to trace the defects of society back to the defects of human nature. The use of symbols in the novel to represent the flaws of human nature, helps create this theme. The conch is one of the main objects the boys use and has