Free Essay

Classification of Differential Equations

In: Science

Submitted By kemuki
Words 336
Pages 2
SUPPLIMENTARY PROBLEMS 1. Write the given equation in standard form
3.16. exy'-x=y' solution exy'-x=y' =exy'-y'=x ex-1y'=x y'=xex-1 ANS

3.25 dy+dx=0 Solution. dy+dx=0 dydx+1=0 y'=-1 2. The differential equations are given in both standard and differential form. Determine whether the equation in standard form are homogeneous and/or linear, if not linear, whether they are Bernoulli: determine whether the equations in differential form as given are separable and/or exact.
3.28 y'=xy+1: xy+1dx-dy=0 Solution. y'=xy+1 = y'-xy=1 this is in the form y'+pxy=qx for px=-x , qx1 thus this is linear next xy+1dx-dy=0 is not separable si9nce the variables cannot be separeted.
To check for exactness take Mx=xy+1 and Nx=-1 for an exact equation, ∂M∂y=∂N∂x
∂M∂y=x and ∂n∂y=0 thus ∂M∂y≠∂N∂x so it is not exact

3.30 y'=x2y2: -x2dx+y2dy=0 Solution. y'=x2y2 =≫ y'+0y=x21y2
The equation is of the form y'+pxy=qxyn where px=0, qx=x2 and n=-2 so it is a Bernoulli equation
For -x2dx+y2dy=0 it is separable since the variables are separated.
To check for exactness take Mx=-x2 and Ny=y2 thus ∂M∂y=0 and ∂N∂x=0 hence its EXACT.
3.35 y'=2xy+x: 2xye-x2+xe-x2dx-e-x2dy=0
For y'=2xy+x =≫y'-2xy=x
This is in the form y'+pxy=qx for px=-2x and qx=x Thus it’s a linear equation
2xye-x2+xe-x2dx-e-x2dy=0 this is not separable since the variables are not separated to check for exactness take Mx= 2xye-x2+xe-x2 and Nx= -e-x2
∂M∂y=2xe-x2 and ∂N∂x=2xe-x2 thus ∂M∂y=∂N∂x and so it is EXACT…...

Similar Documents

Premium Essay

Differential Equation

...Exercise 1 – Find the first derivative and the second derivative of the following functions Answer: Applying constant function and power function rule (A) Y = 3 + 10X + 5X2 dY/dX = 0 + 1.10.X1-1 +2.5.X2-1 dY/dX = 10 + 10X d2Y/ dX2 = 0 + 1.10.X1-1 d2Y/ dX2 = 10 (B) Y = 2X (4 + X3 ) Y = 8X + 2X4 dY/dX = 1.8.X1-1 + 4.2.X4-1 dY/dX = 8 + 8X3 d2Y/ dX2 = 0 + 3.8.X3-1 d2Y/ dX2 = 24X2 (C) Y = 3 /X2 Y = 3X-2 dY/dX = -2.3.X-2-1 dY/dX = -6X-3 dY/dX = -6/X3 d2Y/ dX2 = -3.-6X-3-1 d2Y/ dX2 = 18X-4 d2Y/ dX2 = 18/X4 (D) Y = 18T – 2T2 dY/dT = 1.18.T1-1 – 2.2.T2-1 dY/dT = 18 – 4T d2Y/ dT2 = 0 – 1.4.T1-1 d2Y/ dT2 = - 4 Exercise 2 - Find the partial Derivative of Y with respect to X Answer (A) Y = 10 + 3Z + 2X ∂Y/∂X = 0 + 0 + 1.2.X1-1 ∂Y/∂X = 2 (B) Y= 18Z + X2 + Z.X ∂Y/∂X = 0 + 2.1.X2-1 + Z ∂Y/∂X = 2X + Z Application - The nursing home industry is growing rapidly because the aging of American population. According to the study of an economist, the average cost per patient day of a nursing home can be approximated by C = A – 0.16B + 0.002B2 Where, B is the nursing home’s number of patient days per year ( in thousands) and A is the number that depends on the location and other factors but not on B. Based on the information , how big must a nursing home be ( in terms of patient – days) to minimize the cost per patient day ? Answer – C= ƒ (A, B) Where C is Avg. Cost per patient day A is variable depends on location and other......

Words: 404 - Pages: 2

Premium Essay

Differential Pricing in Telecom Sector

...Differential Pricing: Many important industries involve technologies that exhibit increasing returns to scale, large fixed and sunk costs, and significant economies of scope. Two important examples of such industries are telecommunications services and information services. In each of these cases the relevant technologies involve high fixed costs, significant joint costs and low, or even zero, marginal costs. Setting prices equal to marginal cost will generally not recoup sufficient revenue to cover the fixed costs and the standard economic recommendation of "price at marginal cost" is not economically viable. Some other mechanism for achieving efficient allocation of resources must be found. The outcome of this investigation is that (i) efficient pricing in such environments will typically involve prices that differ across consumers and type of service; (ii) producers will want to engage in product and service differentiation in order for this differential pricing to be feasible; and, (iii)differential pricing will arise naturally as a result of profit seeking by firms. It follows that differential pricing can generally be expected to contribute to economic efficiency Thus differential pricing is “the practice of selling the same product to different customers at different prices even though the cost of sale is the same to each of them. More precisely, it is selling at a price or prices such that the ratio of price to marginal costs is different in different sales” ...

Words: 2467 - Pages: 10

Premium Essay

Differential Staining

...MLT1 Experiment 5/Task 6 Differential Staining There are many different ways to stain bacteria for viewing under a microscope. For the most part these are categorized as “either simple, nonspecific or differential (specific) (LabPaq, p.128). The most common staining method used is Gram staining. Gram staining is a differential or specific method of staining. Gram staining is a way to tell the difference between gram positive and gram negative bacteria. The cell wall of the gram positive bacteria is made up of several “layers of peptidoglygan,” and “techoic acids,” (LabPaq, p. 128). This peptidoglycan is what absorbs the color part of the crystal violet stain causing the gram positive bacterial cell to appear violet colored when viewed with a microscope. In contrast, the cell wall of the gram negative bacterial cell is not as thick as that of the gram positive cell wall. Peptidoglygans are on the inside of the cell rather than in outer layers. The outer part of the gram negative cell is made up of phospholipids and lipipoly-saccharides (Betsy and Keough, 2005). The outer cell wall does not hold onto the violet color of the crystal violet and appears pink in color when viewed under a microscope. The purpose of iodine staining is to aid the bacteria to keep the stain by creating an iodine-crystal violet mixture that will not dissolve. Iodine is also known as a mordant in this case. A mordant is usually an inorganic oxide that when mixed with...

Words: 417 - Pages: 2

Premium Essay


...Markisha Harris Introduction to Corrections Unit 7 Assignment 02 January 2014 Classification The classification process in corrections differs from state to state. According to the Department of Corrections in the State of Montana, classification is basically a way of consolidating and controlling the offenders that are housed in the prison all based upon their identity. Offenders are also categorized by their behavior and other potential risks. When it comes to prison facilities, there are specific purposes of classification and specific ways of how classification systems are used to place prisoners. In the State of Montana, the primary or main purpose of the system of classification in the correctional system is too basically use a method to figure out whether or not which proper encoding and safekeeping step is right for every single convict. A secondary purpose of the classification process in the State of Montana is to allow the prisoners a chance to change their lives around in a positive way. By allowing this opportunity, it will instill protection, safety, and security of other inmates, correctional officers, and the society. The classification system has a goal. The goal is to have order within the correctional facilities and to maintain proper security and safety to everyone that are in contact with offenders. In order for this goal to be achieved, all inmates has to be classified right after the admission process. After the admission process,......

Words: 703 - Pages: 3

Free Essay

Math Equation

...Chapter 2: Equations and Inequalities MAT 1103: Fundamentals of Mathematics 2.1 Equations 1) Equation Statement indicating that 2 quantities are true. Example: Solution set: 3x − 2 = 10 values of variable that satisfy equations. 2) Restricted values a. Fraction: 1 , x≠a x−a b. Radical: c. Logarithmic: x−a, x≥a log( x − a) , x > a 3) Solve Linear equation in 1 variable ax + b = 0 Example 1: Solve a. (a and b are real numbers and a ≠ 0) 2x + 3 = 0 b. 3( x + 2) = 5 x + 2 c. 3y 2 1 − = y 2 3 5 1 Chapter 2: Equations and Inequalities MAT 1103: Fundamentals of Mathematics 4) Solve Rational Equations Example 2: Solve a. 3 7 + =2 5 x+2 b. 3x 5 − =3 x −1 x + 3 2.2 Applications of Linear Equations 1) English-mathematics vocabulary Mathematical operator + x English words More, greater, add, sum, exceeds, increase, higher, total, extra Less, difference, lower, minus, decrease, fewer Times, multiple Mathematical ratio 2x 3x 1/3 x ¼x English words Double Triple One third One quarter 2) General guideline for solving word problem a. Read the problem. b. Read the problem again. c. Draw a picture / table / flow chart. d. Find and label the unknowns that you are looking for. e. Find and label the known quantities. f. Write down all the formulas and relations between the known and unknown. g. Solve the problem. h. Check the answer & reply in words. 2 Chapter 2: Equations and Inequalities MAT 1103:......

Words: 773 - Pages: 4

Free Essay

Gravity Equation

...NBER WORKING PAPER SERIES THE GRAVITY EQUATION IN INTERNATIONAL TRADE: AN EXPLANATION Thomas Chaney Working Paper 19285 NATIONAL BUREAU OF ECONOMIC RESEARCH 1050 Massachusetts Avenue Cambridge, MA 02138 August 2013 I want to thank Fernando Alvarez, Michal Fabinger, Xavier Gabaix, Sam Kortum, Bob Lucas, Jim Tybout, Jon Vogel and seminar participants in Berkeley, Bilkent, Bocconi, Boston University, Chicago, Erasmus, Hitotsubashi, LBS, Louvain-CORE, LSE, the NY Fed, Oxford, Princeton, Rochester, Sciences Po, Toulouse, UBC Vancouver, Yale and Zurich for helpful discussions, and NSF grant SES-1061622 for financial support. I am indebted to Jong Hyun Chung, Stefano Mosso and Adriaan Ten Kate for their research assistance. During the last year, I have received compensation for teaching activities from the Toulouse School of Economics, as well a research grant from the National Science Foundation (SES-1061622), in excess of $10,000. The views expressed herein are those of the author and do not necessarily reflect the views of the National Bureau of Economic Research. NBER working papers are circulated for discussion and comment purposes. They have not been peerreviewed or been subject to the review by the NBER Board of Directors that accompanies official NBER publications. © 2013 by Thomas Chaney. All rights reserved. Short sections of text, not to exceed two paragraphs, may be quoted without explicit permission provided that full......

Words: 18767 - Pages: 76

Premium Essay


...LRBI Checklist Differential Reinforcement Differential Reinforcement ifferential reinforcement is the reinforcement of one form of behavior and not another, or the reinforcement of a response under one condition but not another. Differential reinforcement uses positive reinforcement to differentiate or separate appropriate student behavior from inappropriate behavior by increasing one while decreasing the other. Definition D Things to Do  Select behaviors to be decreased or increased.  Select alternative, incompa tible or communicative behavio , rs to be taught.  Determine time interval.  Set criterion.  Ignore inappropriate behav iors.  Monitor the student’s performance. Level 1: Positive Interaction Procedures 1 LRBI Checklist Differential Reinforcement There are six strategies that comprise or make up differential reinforcement: Select behaviors to be decreased or increased. Select behaviors to be decreased or increased. These behaviors should be objectively defined and must be observable and measurable. For DRO, DRI, DRA, and DRC, there will be a specific behavior to decrease, such as talk-outs, hitting, not following directions, out of seat, disrupting other students, or tantrums. For DRH, there will be a specific behavior to increase, such as contributing in class, positive social interactions, getting to class on time, or task completion. D R H RL D problem behaviors Level 1: Positive Interaction Procedures ...

Words: 1949 - Pages: 8

Free Essay


...Classification has always been the main source of the simplifying all of the information that has been found throughout the span of human thought. Science in its most basic from is nothing more than just a quest of knowledge “to seek what regularities there may be”(Sagan) and the existence of that knowledge is explained by classification(Foucault). Today’s society is dominated by an absurd amount of technologically powered apparatuses like: cell phones, computers, tablets, tables, cars, and the list ventures endlessly forward and like every “small step for a man, and one giant leap forward for mankind” there are repercussions(The Space Race). One of the things that changed with the evolution of technology from its primitive state to its more modern version is news; once only told by an unbiased, or at least strived to be, newscasters like Walter Cronkite(Cronkite) and people were forced to watch the whole thing until what they were really interested in came on has now turned into an extremely selective and biased way of reporting the news with outstandingly big downsides that has lead society to be harshly less informed about what is going on around them and more susceptible to having their opinion swayed with possible misclassification of news. By the end of 2010 more people claimed to be receiving their news intake from the internet than from newspapers(Featured in Social Media) and with the simplified access to......

Words: 1224 - Pages: 5

Premium Essay

Accounting Equation

...There are numerous aspects to the accounting equation and each has its own set of criteria. In order to maintain proper balance of the account equation, assets equaling liabilities plus shareholder’s equity there are several things to consider which include recordable transactions and financial statements. A transaction is any event that has an impact on the financial statements of the business. In order for a transaction to be recorded it must result in assets equaling liabilities plus shareholder’s equity. Examples of recordable transactions include; the sale of merchandise to a customer, a purchase of supplies or equipment, and borrowing funds from a lender. The aforementioned equation assets = liabilities + shareholder’s equity is the fundamental accounting equation. For an account to have a transaction post to it both sides of the equation must remain equal. You could not for example increase assets without subsequently decreasing liabilities or shareholder’s equity. There are four primary financial statements in accounting; Balance Sheet, Income Statement, Statement of Retained Earnings, and Statement of Cash Flows. A Balance Sheet is a statement that shows all of a business’s assets, liabilities, and equity for a point in time. The function of a balance sheet is to show a company’s liquidity and calculate net worth. An Income Statement is a statement that measures a company’s financial performance over a given period, a year for example. The primary function of......

Words: 372 - Pages: 2

Premium Essay


...Different gas shoppers ITT Tech Institute Classifications There are many types of people who purchase gasoline, but three types are easy to spot. All three have an obvious physical appearance which sets them apart from other gas purchasers. Each also has a distinctive pumping technique and freely vocalizes his opinion. Finally, each type has a particular way of annoying every other patron at the same location. The first type of gas purchaser is Leisure Lou. He can be identified immediately by his appearance. He drives a station wagon with wooden side panels and a bumper sticker that tells the world that he is retired. His Hawaiian print shirt captures everyone's attention. He wears Bermuda shorts and black socks with sandals. On his head sits a fly fisherman's hat which is his obvious display of a continuous state of leisure. His pumping technique also sets him apart from the normal crowd. When he parks his car, he stops at the first pump, preventing the use of the pump in front of him. He always pays for his gas in advance, so that the attendant can preset the pump and he will not have to worry about stopping the pump himself. He activates the pump before opening the access door and unscrewing the tank lid. He tucks the hose under his elbow and spills gas on his hip while opening the tank. As Leisure Lou leans against the side of his car, he pumps his gas absent-mindedly, confident that the pump will stop before gas pours onto the ground. Once the pump stops, he......

Words: 919 - Pages: 4

Free Essay

Ordinary Differential Equations

...MATH 364A: Ordinary Differential Equations (Midterm 1) Name: Student ID: Signature: Question 1 (40 points) Solve the following initial value problems. (a) y + t3 y = t3 y(0) = 0. (b) y = − (1+x) y y(−1) = 1. Question 2 (40 points) Solve the second-order initial value problem 2y − 3y − 5y = 0 y(0) = 0 2 y (0) = 1. Question 3 (40 points) For each equation below, first determine whether the equation is exact or not exact. If the equation is exact, find the solution. (a) cos(y) + (2y − x sin(y)) dy = 0. dx (b) 3xy 2 + (y 3 + 3x2 ) 3 dy = 0. dx Question 4 (40 points) Consider the following first-order differential equation y = (y 2 − 1)(4 − y 2 ). Find all critical (equilibrium) solutions and classify their stability. 4 Question 5 (40 points) (True/False) In each of the following, determine whether the given function solves the given differential equation or initial value problem. (a) If k > 0 denotes any real constant, then the function y(t) = e−kt solves the initial value problem y = −ky y(0) = 1. (b) The function y(t) = (1 − t)−1 = 1/(1 − t) solves the initial value problem y = y2 y(0) = 1. (c) The function y(t) = e2t solves the second-order initial value problem y = 2y y(0) = 1 y (0) = 2. (d) Any function y(x) defined by the implicit equation x4 + 2x2 y 2 + y 4 = C solves the differential equation x3 + xy 2 + (x2 y + y 3 )y = 0. 5 Question 6 (Extra Credit, 10 points) An object with......

Words: 324 - Pages: 2

Free Essay

Differential Gear

...The differential was first invented in China, in the third century,A.D. After turning a car the outside wheel has to turn faster then the inside one in order to cover the greater distance. That is why two wheels are not driven at a same speed. So we need a differential gear. A car differential is a place halfway between the wheels, on either the front or both axes. Main mechanisms: Wheels receive power from the motor via drive shaft. The power receiving wheels which makes the vehicle move fowrord are called as the drive wheels. Wheels receive power from the motor via drive shaft. The main function of the differential gear is to allow the drive wheels to turn at different Rpm while from the engine. The main characterics of differential is to, * To aim the engine power at wheels. * To transmit the power to the wheels while allowing them to rotate at differential speeds. the left wheel has to travel more distance compared to the right wheel. If the wheels where connected using a soil shaft. The wheels where connected using a soil shaft. The wheels would have to slip to accomplish the turn. The mechanism in a differential allows left and right wheels to turn at different Rpm, while transferring power to both the wheels. Differential parts: There are three types of gear available, Pinion drive gear: Transfers power from the drive shaft to the ring gear. Ring gear: Transfers power to the differential case assembly Side/Spider gears: Helps both wheels to......

Words: 656 - Pages: 3

Free Essay


...stop to take a bathroom break then they’ll miss an opportunity to drop their delivery; and as everyone knows time is money. So in other words, they’ll “hold it” until they explode. Usually when the explosion goes off, it’s in their pants. When they finally pull over to handle the situation they’ll proudly climb out of their truck with their head held high and ask for a shower. Meanwhile, the poop turned to liquid from sitting on it for so long; so when the driver proceeds with their “walk of shame” it drips on to the floor leaving a trail. It is a very disgusting cite. Even with the distasteful hygiene the stinky fat-o possess, they uphold the same mind set as the professional clean cuts, which is enjoyable. All of these different classifications of professional truck drivers vary in appearance and attitude, but they definitely have one thing in common and that’s their career. Now, it’s pretty obvious that all of these drivers need some type of change in their life’s, some more than others. Maybe one day the creepy flirts that roam the aisles at Pilot Travel Center will finally find somebody interested in them; perhaps Curtis will discard those cheesy pickup lines and find a clever phrase instead. Possibly the professional clean cuts will find a job that they are better qualified for. Perchance the grumpy jerks will cure their internal issues and seek counseling or anger management; in addition to that guidance maybe those jerks will learn respect and not take out their......

Words: 2067 - Pages: 9

Free Essay

Differential Equations

...Differential Equations by tarungehlots Concepts of Differential Equation Consider a variable that might denote the per capita capital stock level in an economy. When convenient, we can recognize that the level of capital depends upon time by addition the time argument. Doing so, we would write the per capita capital stock level as , rather than just writing . If we think of time as unfolding continuously, we can also think of capital as being a continuous function of time, and we can assume that the function has derivate . This derivative is the instantaneous change in per capita capital. In many presentations, is written as , but we will here use the more familiar notation . By writing the derivate as , so that we include the time argument, we are emphasizing that the value of the derivative may change with time. Because it is typically cumbersome to repeatedly write down the argument, we can also write the derivative as just , while remembering that its value may change with time. A differential equation is an equation that relates the time derivative of a variable to its level. An example is the equation (1) . The variable is called a state variable because it gives the state of the system at any given point in time. In our example, gives the state or level of per capita capital stock. 2. A Dynamic System The basic dynamic principle is the idea that “the way things are determines the......

Words: 2717 - Pages: 11

Free Essay

Rent Differentials

... KOFORIDUA POLYTECHNIC RENT DIFFERENTIALS BETWEEN RESIDENTIAL AREAS AND ZONGO COMMUNITIES IN GHANA AND DETERMINANTS OR FACTORS THAT MOTIVATE PEOPLE TO STAY OR HIRE AT SUCH PLACES/ AREAS. A CASE STUDY OF KOFORIDUA OLD ESTATES AND KOFORIDUA ZONGO. BY BANAFO BENJAMIN DUODU OBED FEKOW SARPONG ALFRED ANGMOR SOLOMON K. OPOKU MENSAH A. RICHARD YEBOAH A. DERRICK ANUM AYIVOR ISHMAEL E. OSABUTEY CATHERINE A. OWUSU BISMARK 2012 DEDICTION Our expectation is that this case study will be dedicated to all potential researchers who might be embarking on this same case study and to Mr. Jamal Mohammed paved the way for us to embark on this practical research. ACKNOWLEDGEMENT We are most grateful to the Lord Almighty for strength, knowledge and wisdom granted to us in our area of study. We also acknowledge every member of our group who contributed their effort to this successful research. TABLE OF CONTENT PAGE CHAPTER ONE * BACKGROUND INFORMATION ...

Words: 2429 - Pages: 10

Kim Kardashian - Love Magazine Spring - Summer 2015 | Sora no Otoshimono: Tokeijikake no Angeloid | (Ambient, Drone, Electroacou...