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Words 2717

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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 way things change.” A differential equation is one way of modeling the basic dynamic principle. The state variable indicates the way things are, while the time derivative indicates how things change. The differential equation itself is what relates the two. More precisely, a differential equation typically presents a functional relationship, showing how depends upon . When we want to…...

...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......

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... Accounting Equation Franklin Weatherspoon ACC/300 OCTOBER 26, 2013 Dr. M Moczynski, CPA, CGFM Accounting Equation In the business world an individual cannot understand a balance sheet, income statement nor transaction recordings within a general ledger, until he or she understands the basic accounting equation: Assets = Liabilities + Owners Equity (Barclay, 2013). Following is a discussion on the relationship of accounting equation and the components of a balance sheet; along with examples showing the affect of each (University of Phoenix, 2013). Relation in Components The accounting formula is the way double-entry bookkeeping is formulated. The accounting formula, which is known as the balance sheet equation, signifies the connection between the assets, liabilities, and owner's equity of a small company (Peeler, 2013). The accounting formula basically expresses a company’s assets, which is obtained either by liabilities or by the company’s capital. The equation has to balance since the company’s entirety, which is known as assets are brought with something like liability or the company’s capital (Peeler, 2013). The effect of one another The company’s equity is affected by capital, functioning as stock. The company has income, which is revenue minus expenses, and gains minus loses, and sometimes extra capital and withdrawals are known as dividends. When the closing of the month approaches, each item will......

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...associated with long-term or fixed volume/capacity agreements; collateral benefits provided by the access seeker (for example through stimulating extra demand for the access 20 21 11 However, differential pricing can reduce efficient competition. Preferential access pricing between a limited group of network operators can have the effect of discouraging entry of more efficient operators.22 Differential pricing can also discourage investment. In an industry where assets often have little alternative use, there is scope for an access provider to appropriate the commercial returns to the assets 23 of access seekers through high access prices. There appears to be even greater scope for differential pricing to reduce efficient competition where an access provider provides preferential pricing to its own vertically-integrated operations or to its subsidiaries or associates. The incentive for the access provider to discriminate against competitors can inhibit efficient entry and competition in those markets. The Commission expects that in most undertakings the same menu of offerings will be available to all access seekers on a non-discriminatory basis. Where an undertaking provides scope for differential pricing not based on costs the Commission must be satisfied that such differential pricing will promote competition and will enhance the efficient use of, and investment in, infrastructure. As discussed below, when arbitrating disputes the Commission will use a......

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...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” ...

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...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...

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...Action Equation Michele Riazzi Grand Canyon University/LDR645 May 27, 2014 Action Equation Plan Do: What do you need employees to do? * Employees must enroll a minimum of 5 students * It is preferable for students to begin closer to the beginning of the month * Enrollment counselors should be listening by asking open ended questions * Direct students to take an action by beginning the application * Ask for referrals each month with new and existing students * Try to have all students for the month registered by the 15th to begin the next month Know: What do employees need to know to take action? * Know what to do and why we do it * Enrollment Counselors need to understand how the revenue stream is affected by monthly expectations. * Database management is critical to create the most opportunities with the inquiries provided. * Enrollment Counselors need to set clear expectations with student from the initial conversation. They, students and enrollment counselors, must follow through on the enrollment process. * Enrollment counselors must ask all students and potential students for referrals. This may be accomplished by speaking with current students on a weekly basis. Feel: What do employees need to feel to take action? * Enrollment counselors are usually the first impression potential students have of Grand Canyon University. This is a valuable role and must be taken seriously. * The enrollment counselor’s...

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...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:......

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...NBER WORKING PAPER SERIES THE GRAVITY EQUATION IN INTERNATIONAL TRADE: AN EXPLANATION Thomas Chaney Working Paper 19285 http://www.nber.org/papers/w19285 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......

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...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 ...

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...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......

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...CLASSIFICATION OF FIRST ORDER DIFFERENTIAL EQUATIONS 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 NEXT 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 Solution. 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 NEXT 2xye-x2+xe-x2dx-e-x2dy=0 this is not separable since the variables...

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...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......

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...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......

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...accounting equation, using 100 to 150 words. There are many different rules, regulations and requirements in accounting. However it does not matter how complex an area of accounting may look because accounting is based on one elemental principal which is Assets = Liability + Owners (shareholder) Equity. The key to remember is in any transaction there is always at least two sides and each side of the equation always stays balanced with each other. An asset is something in value a company owns. Examples would be Assets make up cash, property, office equipment, inventory and accounts receivable. Liabilities are existing debts and obligations that are owed by a company. Examples would be salaries, finaning a purchase, notes payable. Owners equity is the owners right to the assets of an entity after all liabilities are paid. Examples would be retained stock, cash invested in the business and paid in capital. Even though there are many different rules, regulations and requirements in accounting. These are the fundamental principals of accounting. ❖ Complete E1-5 on p. 34 of Financial Accounting Assets Cash Cleaning Equipment Cleaning Supplies Accounts Receivable Liability Notes Payable Salaries Payable Accounts Payable Owner Equity Commerce Stock |CheckPoint 2 |10.0 |10.0......

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... 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 ...

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