Three major steps in modeling: Construction of the Model Analysis of the Model Comparison with Experiment or Observation
|
|
- Shawn Webster
- 5 years ago
- Views:
Transcription
1 Section 2.3 Modeling : Key Terms: Three major steps in modeling: Construction of the Model Analysis of the Model Comparison with Experiment or Observation Mixing Problems Population Example Continuous Compounding
2 Mixing Mixing problems involve the mixing of substances and have particular significance to chemistry and biology. The idea is to build a model that predicts the amount of a substance (salt, drugs, etc) in a container. The substance flows into the container at some given rate (input rate), is mixed with the ingredients in the container, and then follows out of the container at some given rate (output rate). We assume the container is thoroughly mixed. We will assume that the substance is neither created nor destroyed in the container. Thus variations in the amount of the substance are due solely to the flows in and out of the tank. If we denote the amount of substance in the container at time t by Q(t), the rate of change of Q, dq/dt, is equal to the rate at which the substance flows in minus the rate at which the substance flows out. dq rate in of substance - rate out of substance dt
3 A usual situation is that we are dealing with a fluid. An amount of substance is dissolved in the fluid that comes into the container, is mixed with the fluid that is in the container (which may contain some of the substance), and then the fluid that exits the container also contains some of the substance dissolved in the fluid. Lets consider a general situation: Let Q(t) be the number of pounds of salt in a tank at time t measured in minutes. Then dq/dt has units the number of pounds of salt per minute. Since we are dealing with a fluid we have so many gallons/minute coming in and so many gallons/minute going out. But we must keep in mind that dq rate in of substance - rate out of substance dt has units pounds/minute.
4 dq rate in of substance - rate out of substance dt We can determine the expression for the RATE IN and the RATE OUT of the substance as follows: RATE IN = (Concentration In) times (Fluid Flow Rate In) (lb/min) (lb/gal) (gal/min) RATE OUT = (Concentration Out) times (Fluid Flow Rate Out) (lb/min) (lb/gal) (gal/min)
5 Example: At time t = 0 a tank contains Q 0 Ib of salt dissolved in 100 gal of water. Assume that water containing 1/4 Ib of salt/gal is entering the tank at a rate of r gal/min and that the well-stirred mixture is draining from the tank at the same rate. (a) Construct the IVP that describes Q the amount of salt in the tank at any time t. Find the amount of salt Q(t) in the tank at any time. (b) Find the limiting amount Q L, that is present after a very long time. RATE IN = (Concentration In) times (Fluid Flow Rate In) = ¼ lb/gal r gal/min = ¼ r lb/min RATE OUT = (Concentration Out) times (Fluid Flow Rate Out) amt of salt in tank at time t gal = r number of gallons in tank at time t min IVP = (Q/100) lb/gal r gal/min = (rq/100) lb/min
6 IVP Is the DE is 1 St order linear? Is it separable? Is it 1st order linear autonomous? Rewriting as a first order linear DE we get It is also 1 st order linear autonomous. In standard form we have For this DE we have the form dy ay dt dq r r r r Q so a, b dt Then the solution is r/ 4 Q Ce 25 Ce r/ 100 Applying the initial condition we find C = Q So the solution of the IVP is ( r / 100) t ( r / 100) t b 100 Q 25 Q 25 e ( r / ) t For completeness the next page shows the work using the1st order linear technique. 0
7 Rewriting as a first order linear DE we get Here is the work for 1 st order linear. Integrating factor is The general solution of the DE is obtained from the following steps. Solving for Q we get Applying the initial condition we find C = Q So the solution of the IVP is
8 (b) Find the limiting amount Q L ; that is, amount present after a very long time. As t we have Q 25; that is, Q L = 25. Let s look at the solution of the IVP closely: The larger the value of r the faster Q 25. Rewrite the solution as The amount of salt in the tank due to the flow processes. Portion of the original salt that remains in the tank at time t.
9 Let s modify things a bit: set r = 3 gal/min and Q 0 = 50 lbs. In this case the solution of the IVP is Find the time T at which Q is within 2% of Q L = 25. (recall that Q(t) > 25) 2% of 25 is 0.5 so we want T when Q = 25.5 Solve for t Next let s find the rate r so that the time T when Q(t) is within 2% of 25 lbs does not exceed 45 min. Here we use We set t = 45, Q = 25.5 and solve for r. We have
10 Although this particular example has no special significance, models of this kind are often used in problems involving a pollutant in a lake, or a drug in an organ of the body, for example, rather than a tank of salt water. In such cases the flow rates may not be easy to determine or may vary with time. Similarly, the concentration may be far from uniform in some cases. Finally, the fluid rates of inflow and outflow may be different, which means that the variation of the amount of liquid in the problem must also be taken into account.
11 Example: A 1000 gallon tank contains 400 gallons of pure water. A valve is opened so that fluid containing 2lbs of salt per gallon enters the tank at the rate of 4 gallons per minute and at the same time a drain valve is opened so fluid exits the tank at 2 gallons per minute. Construct an IVP for this situation. dq rate in of substance - rate out of substance dt 4 gal / min (2lbs / gal) dq dt Q(t) 2gal / min t Q(t) = 4 gal / min (2lbs / gal) -2gal / min, Q(0) = t dq dt Q(t) = 8-2, Q(0) = t As time goes on what happens to the tank in this case? Is the DE is 1 St order linear? Is it separable? Is it 1st order linear autonomous? How do you find the time that this occurs? Do you need to solve the IVP?
12 Now solve the IVP. dq dt Q(t) = 8-2, Q(0) = t dq dt Put the DE into standard form for 1 st order linear: Then p(t) 2 Q(t) t 2 Q(t) dt ln 400+2t so the integrating factor is μ(t) e 400+2t = e = t Q(t) 2 = t Applying theory for 1 st order linear DEs we multiply both sides of the standard form by the integrating factor. d Q μ(t) = t dt Integrate both sides: 2 Q μ(t) = t dt = t + C 2 Solve for Q: t Q = + C t = t + C t t Apply initial condition to get C: C = Solution of IVP: Q = t t
13 SALT t (400+2 t) TIME How do you find the time that the tank contains 800 pounds of salt? Q = t t -1
14 A Population Model The population of mosquitoes in a certain area increases at a rate proportional to the current population, and in the absence of other factors, the population doubles each week. There are 200,000 mosquitoes in the area initially, and predators (birds, bats, and so forth) eat 20,000 mosquitoes/day. Determine the population of mosquitoes in the area at any time. Note that we have two time frames in the information given; weeks and days. Since eventually we want the population after predation lets choose time in days. The population doubles every seven days (ignoring predation) so if P(t) is the population function we have IVP dp rp, P( 0 ) 200, 000 and P( 7 ) 2 P( 0 ) dt Note this is equivalent to 1 week. Solving the IVP for P we get P= Ce rt. Applying P(0) = 200,000 we have C = 200,000. So we have P = 200,000 e rt Next using P(7) = 2P(0) so we can 400, , 000 e 7r solve for r. ln( 2) Using logs we get r 7 Now we are ready to include the predation.
15 We were told that predators (birds, bats, and so forth) eat 20,000 mosquitoes/day. Determine the population of mosquitoes in the area at any time. The IVP here is rate of change = input of mosquitoes output of mosquitoes dp ln( 2) rp 20, 000 P 20, 000, P( 0) 200, 000 dt 7 The DE is first order linear autonomous with a = ln(2)/7 and b = 20,000 so the solution is b at 20, 000 P(t) Ce Ce (ln( 2)/ 7)t a ln( 2) / 7 Applying the initial condition P(0) = 200,000 we find that 140, 000 C 200, ln( 2) ln( 2) t P(t) e 7 201, The next slide is the details for the same computation usin first order linear and integrating factors.
16 We were told that predators (birds, bats, and so forth) eat 20,000 mosquitoes/day. Determine the population of mosquitoes in the area at any time. The IVP here is rate of change = input of mosquitoes output of mosquitoes dp ln( 2) rp 20, 000 P 20, 000, P( 0) 200, 000 dt 7 2 This is a first order linear DE. dp ln( ) P 20, 000 dt 7 ln( 2) ln( 2) Integrating factor: dt t (t) e 7 e 7 But it is also 1 st order linear autonomous. Applying the initial condition P(0) = 200,000 we find that So we have ln( 2) t P(t) e 7 201, , 000 C 200, ln( 2)
17 Continuous Compounding Suppose that a sum of money is deposited in a bank or money fund that pays interest at an annual rate r. The value S(t) of the investment at any time t depends on the frequency with which interest is compounded as well as on the interest rate. Financial institutions have various policies concerning compounding: some compound monthly, some weekly, some even daily. In general, if interest is compounded m times per year, then for an initial deposit S(0) = S 0 the amount in the account at time t is given by S(t) S0 1 m r mt where m is the number of times the interest is compounded in a year. If we assume that compounding takes place continuously, then we can set up a simple initial value problem that describes the growth of the investment.
18 COMPARISON: Suppose $1000 is deposited into an account yielding 5% interest compounded at the following frequencies. How much money is in the account after 5 years? S( t) S 0 1 r m m5 Annually.05 5 S(5) ( 1.05) 1 5 $ Semiannually S(5) (1.025) 2 10 $ Quarterly S(5) (1.0125) 4 $ Monthly Daily compounding gives $ S(5) ( ) 12 $
19 If we assume that compounding takes place continuously, then we can set up a simple initial value problem that describes the growth of the investment. Here we use a bit of calculus: mt r lim S(t) lim S rt 01 S0e m m m This formula for continuous compounding of interest is the same as the solution of the IVP ds = rs, S(0) = S0 dt Returning to our Comparison, where $1000 is invested in an account with an interest rate of 5% compounded continuously. How much money will there be in the account after 5 years? (assume no withdrawals) S(t)= $1000 e.05(5) = $ Recall: compounding monthly gave $
20 In the case of continuous compounding, let us suppose that there may be deposits or withdrawals in addition to the accrual of interest. If we assume that the deposits or withdrawals take place at a constant rate (amount) k, then we have that IVP ds rs, S(0) S0 dt is replaced by ds rs k, S(0) S0 dt Also 1 st order linear or, in standard form for 1 st autonomous. order linear, ds rs k, S(0) S0 dt where k is positive for deposits and k negative for withdrawals. This DE is first order linear and its general solution is rt k S(t) Ce, C an arbitrary constant r Appling the initial condition we get the solution (after a rearrangement) of the IVP to be rt k S(t) S rt 0e e 1 r Return on initial deposit Effect of additional deposit or withdrawal.
21 The advantage of stating the problem in this general way without specific values for S 0, r, or k lies in the generality of the resulting formula S(t). rt k S(t) S rt 0e e 1 r With this formula we can readily compare the results of different investment programs or different rates of return. For instance, suppose that one opens an individual retirement account (IRA) at age 25 and makes annual investments of K = $2000 thereafter in a continuous manner. Assuming a rate of return of 8%, what will be the balance in the IRA at age 65? We have S 0 = 0, r = 0.08, and k = $2000. Then S(40) =(25,000)(e 3.2 1) = $588,313 It is interesting to note that the total amount invested is $80,000, so the remaining amount of $508,313 results from the accumulated return on the investment. The balance after 40 years is also fairly sensitive to the assumed rate. For instance, S(40) = $508,948 if r = 0.075, S(40) = $98,36494 if r Alternatively you can have an IRA that is tied to the stock market behavior, rather than a fixed rate of interest. In such a case this formula is no applicable.
Section 2.5 Mixing Problems. Key Terms: Tanks Mixing problems Input rate Output rate Volume rates Concentration
Section 2.5 Mixing Problems Key Terms: Tanks Mixing problems Input rate Output rate Volume rates Concentration The problems we will discuss are called mixing problems. They employ tanks and other receptacles
More informationChapter 2: First Order ODE 2.4 Examples of such ODE Mo
Chapter 2: First Order ODE 2.4 Examples of such ODE Models 28 January 2018 First Order ODE Read Only Section! We recall the general form of the First Order DEs (FODE): dy = f (t, y) (1) dt where f (t,
More informationCh 2.3: Modeling with First Order Equations. Mathematical models characterize physical systems, often using differential equations.
Ch 2.3: Modeling with First Order Equations Mathematical models characterize physical systems, often using differential equations. Model Construction: Translating physical situation into mathematical terms.
More informationSect2.1. Any linear equation:
Sect2.1. Any linear equation: Divide a 0 (t) on both sides a 0 (t) dt +a 1(t)y = g(t) dt + a 1(t) a 0 (t) y = g(t) a 0 (t) Choose p(t) = a 1(t) a 0 (t) Rewrite it in standard form and ḡ(t) = g(t) a 0 (t)
More informationModeling with First Order ODEs (cont). Existence and Uniqueness of Solutions to First Order Linear IVP. Second Order ODEs
Modeling with First Order ODEs (cont). Existence and Uniqueness of Solutions to First Order Linear IVP. Second Order ODEs September 18 22, 2017 Mixing Problem Yuliya Gorb Example: A tank with a capacity
More informationHomework 2 Solutions Math 307 Summer 17
Homework 2 Solutions Math 307 Summer 17 July 8, 2017 Section 2.3 Problem 4. A tank with capacity of 500 gallons originally contains 200 gallons of water with 100 pounds of salt in solution. Water containing
More informationP (t) = rp (t) 22, 000, 000 = 20, 000, 000 e 10r = e 10r. ln( ) = 10r 10 ) 10. = r. 10 t. P (30) = 20, 000, 000 e
APPM 360 Week Recitation Solutions September 18 01 1. The population of a country is growing at a rate that is proportional to the population of the country. The population in 1990 was 0 million and in
More information4. Some Applications of first order linear differential
September 9, 2012 4-1 4. Some Applications of first order linear differential Equations The modeling problem There are several steps required for modeling scientific phenomena 1. Data collection (experimentation)
More informationMath 2Z03 - Tutorial # 3. Sept. 28th, 29th, 30th, 2015
Math 2Z03 - Tutorial # 3 Sept. 28th, 29th, 30th, 2015 Tutorial Info: Tutorial Website: http://ms.mcmaster.ca/ dedieula/2z03.html Office Hours: Mondays 3pm - 5pm (in the Math Help Centre) Tutorial #3: 2.8
More informationMA 214 Calculus IV (Spring 2016) Section 2. Homework Assignment 2 Solutions
MA 214 Calculus IV (Spring 2016) Section 2 Homework Assignment 2 Solutions 1 Boyce and DiPrima, p 60, Problem 2 Solution: Let M(t) be the mass (in grams) of salt in the tank after t minutes The initial-value
More informationName: October 24, 2014 ID Number: Fall Midterm I. Number Total Points Points Obtained Total 40
Math 307O: Introduction to Differential Equations Name: October 24, 204 ID Number: Fall 204 Midterm I Number Total Points Points Obtained 0 2 0 3 0 4 0 Total 40 Instructions.. Show all your work and box
More informationFirst Order ODEs, Part II
Craig J. Sutton craig.j.sutton@dartmouth.edu Department of Mathematics Dartmouth College Math 23 Differential Equations Winter 2013 Outline Existence & Uniqueness Theorems 1 Existence & Uniqueness Theorems
More informationLecture Notes for Math 251: ODE and PDE. Lecture 6: 2.3 Modeling With First Order Equations
Lecture Notes for Math 251: ODE and PDE. Lecture 6: 2.3 Modeling With First Order Equations Shawn D. Ryan Spring 2012 1 Modeling With First Order Equations Last Time: We solved separable ODEs and now we
More informationLesson 10 MA Nick Egbert
Overview There is no new material for this lesson, we just apply our knowledge from the previous lesson to some (admittedly complicated) word problems. Recall that given a first-order linear differential
More informationMath 392 Exam 1 Solutions Fall (10 pts) Find the general solution to the differential equation dy dt = 1
Math 392 Exam 1 Solutions Fall 20104 1. (10 pts) Find the general solution to the differential equation = 1 y 2 t + 4ty = 1 t(y 2 + 4y). Hence (y 2 + 4y) = t y3 3 + 2y2 = ln t + c. 2. (8 pts) Perform Euler
More informationOrdinary Differential Equations
Ordinary Differential Equations Swaroop Nandan Bora swaroop@iitg.ernet.in Department of Mathematics Indian Institute of Technology Guwahati Guwahati-781039 A first-order differential equation is an equation
More informationName: Solutions Final Exam
Instructions. Answer each of the questions on your own paper. Put your name on each page of your paper. Be sure to show your work so that partial credit can be adequately assessed. Credit will not be given
More informationLECTURE 4-1 INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS
130 LECTURE 4-1 INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS DIFFERENTIAL EQUATIONS: A differential equation (DE) is an equation involving an unknown function and one or more of its derivatives. A differential
More informationDifferential equations
Differential equations Math 27 Spring 2008 In-term exam February 5th. Solutions This exam contains fourteen problems numbered through 4. Problems 3 are multiple choice problems, which each count 6% of
More informationof 8 28/11/ :25
Paul's Online Math Notes Home Content Chapter/Section Downloads Misc Links Site Help Contact Me Differential Equations (Notes) / First Order DE`s / Modeling with First Order DE's [Notes] Differential Equations
More informationElementary Differential Equations
Elementary Differential Equations George Voutsadakis 1 1 Mathematics and Computer Science Lake Superior State University LSSU Math 310 George Voutsadakis (LSSU) Differential Equations January 2014 1 /
More informationFirst Order Differential Equations Chapter 1
First Order Differential Equations Chapter 1 Doreen De Leon Department of Mathematics, California State University, Fresno 1 Differential Equations and Mathematical Models Section 1.1 Definitions: An equation
More informationPractice Midterm 1 Solutions Written by Victoria Kala July 10, 2017
Practice Midterm 1 Solutions Written by Victoria Kala July 10, 2017 1. Use the slope field plotter link in Gauchospace to check your solution. 2. (a) Not linear because of the y 2 sin x term (b) Not linear
More informationFirst Order Linear DEs, Applications
Week #2 : First Order Linear DEs, Applications Goals: Classify first-order differential equations. Solve first-order linear differential equations. Use first-order linear DEs as models in application problems.
More informationMATH 2410 Review of Mixing Problems
MATH 2410 Review of Mixing Problems David Nichols The following examples explore two different kinds of mixing problems. The word problems are very similar, but the differential equations that result are
More informationMath 308 Exam I Practice Problems
Math 308 Exam I Practice Problems This review should not be used as your sole source for preparation for the exam. You should also re-work all examples given in lecture and all suggested homework problems..
More informationMath 308 Exam I Practice Problems
Math 308 Exam I Practice Problems This review should not be used as your sole source of preparation for the exam. You should also re-work all examples given in lecture and all suggested homework problems..
More informationMATH 251 Examination I October 10, 2013 FORM A. Name: Student Number: Section:
MATH 251 Examination I October 10, 2013 FORM A Name: Student Number: Section: This exam has 13 questions for a total of 100 points. Show all you your work! In order to obtain full credit for partial credit
More informationFirst Order Linear DEs, Applications
Week #2 : First Order Linear DEs, Applications Goals: Classify first-order differential equations. Solve first-order linear differential equations. Use first-order linear DEs as models in application problems.
More informationIt is convenient to think that solutions of differential equations consist of a family of functions (just like indefinite integrals ).
Section 1.1 Direction Fields Key Terms/Ideas: Mathematical model Geometric behavior of solutions without solving the model using calculus Graphical description using direction fields Equilibrium solution
More informationDEplot(D(y)(x)=2*sin(x*y(x)),y(x),x=-2..2,[[y(1)=1]],y=-5..5)
Project #1 Math 181 Name: Email your project to ftran@mtsac.edu with your full name and class on the subject line of the email. Do not turn in a hardcopy of your project. Step 1: Initialize the program:
More informationModeling with first order equations (Sect. 2.3). The mathematical modeling of natural processes.
Modeling with first order equations (Sect. 2.3). The mathematical modeling of natural processes. Main example: Salt in a water tank. The experimental device. The main equations. Analysis of the mathematical
More informationLast quiz Comments. ! F '(t) dt = F(b) " F(a) #1: State the fundamental theorem of calculus version I or II. Version I : Version II :
Last quiz Comments #1: State the fundamental theorem of calculus version I or II. Version I : b! F '(t) dt = F(b) " F(a) a Version II : x F( x) =! f ( t) dt F '( x) = f ( x) a Comments of last quiz #1:
More informationMath Spring 2014 Homework 2 solution
Math 3-00 Spring 04 Homework solution.3/5 A tank initially contains 0 lb of salt in gal of weater. A salt solution flows into the tank at 3 gal/min and well-stirred out at the same rate. Inflow salt concentration
More informationChapter1. Ordinary Differential Equations
Chapter1. Ordinary Differential Equations In the sciences and engineering, mathematical models are developed to aid in the understanding of physical phenomena. These models often yield an equation that
More informationSolutions to the Review Questions
Solutions to the Review Questions Short Answer/True or False. True or False, and explain: (a) If y = y + 2t, then 0 = y + 2t is an equilibrium solution. False: (a) Equilibrium solutions are only defined
More informationSolutions to the Review Questions
Solutions to the Review Questions Short Answer/True or False. True or False, and explain: (a) If y = y + 2t, then 0 = y + 2t is an equilibrium solution. False: This is an isocline associated with a slope
More informationMATH 251 Examination I October 8, 2015 FORM A. Name: Student Number: Section:
MATH 251 Examination I October 8, 2015 FORM A Name: Student Number: Section: This exam has 14 questions for a total of 100 points. Show all you your work! In order to obtain full credit for partial credit
More informationChapter 2 Notes, Kohler & Johnson 2e
Contents 2 First Order Differential Equations 2 2.1 First Order Equations - Existence and Uniqueness Theorems......... 2 2.2 Linear First Order Differential Equations.................... 5 2.2.1 First
More informationProblem Set. Assignment #1. Math 3350, Spring Feb. 6, 2004 ANSWERS
Problem Set Assignment #1 Math 3350, Spring 2004 Feb. 6, 2004 ANSWERS i Problem 1. [Section 1.4, Problem 4] A rocket is shot straight up. During the initial stages of flight is has acceleration 7t m /s
More informationSolutions to Math 53 First Exam April 20, 2010
Solutions to Math 53 First Exam April 0, 00. (5 points) Match the direction fields below with their differential equations. Also indicate which two equations do not have matches. No justification is necessary.
More information4. Some Applications of first order linear differential
August 30, 2011 4-1 4. Some Applications of fist ode linea diffeential Equations The modeling poblem Thee ae seveal steps equied fo modeling scientific phenomena 1. Data collection (expeimentation) Given
More informationThe Fundamental Theorem of Calculus: Suppose f continuous on [a, b]. 1.) If G(x) = x. f(t)dt = F (b) F (a) where F is any antiderivative
1 Calulus pre-requisites you must know. Derivative = slope of tangent line = rate. Integral = area between curve and x-axis (where area can be negative). The Fundamental Theorem of Calculus: Suppose f
More informationMATH 312 Section 3.1: Linear Models
MATH 312 Section 3.1: Linear Models Prof. Jonathan Duncan Walla Walla College Spring Quarter, 2007 Outline 1 Population Growth 2 Newton s Law of Cooling 3 Kepler s Law Second Law of Planetary Motion 4
More informationWill Murray s Differential Equations, IV. Applications, modeling, and word problems1
Will Murray s Differential Equations, IV. Applications, modeling, and word problems1 IV. Applications, modeling, and word problems Lesson Overview Mixing: Smoke flows into the room; evenly mixed air flows
More information8.a: Integrating Factors in Differential Equations. y = 5y + t (2)
8.a: Integrating Factors in Differential Equations 0.0.1 Basics of Integrating Factors Until now we have dealt with separable differential equations. Net we will focus on a more specific type of differential
More informationReview Problems for Exam 2
Calculus II Math - Fall 4 Name: Review Problems for Eam In question -6, write a differential equation modeling the given situations, you do not need to solve it.. The rate of change of a population P is
More informationDifferential Equations Class Notes
Differential Equations Class Notes Dan Wysocki Spring 213 Contents 1 Introduction 2 2 Classification of Differential Equations 6 2.1 Linear vs. Non-Linear.................................. 7 2.2 Seperable
More informationSec 2.3 Modeling with First Order Equations
Sc.3 Modling with First Ordr Equations Mathmatical modls charactriz physical systms, oftn using diffrntial quations. Modl Construction: Translating physical situation into mathmatical trms. Clarly stat
More informationLinear Variable coefficient equations (Sect. 2.1) Review: Linear constant coefficient equations
Linear Variable coefficient equations (Sect. 2.1) Review: Linear constant coefficient equations. The Initial Value Problem. Linear variable coefficients equations. The Bernoulli equation: A nonlinear equation.
More informationMAT 275 Test 1 SOLUTIONS, FORM A
MAT 75 Test SOLUTIONS, FORM A The differential equation xy e x y + y 3 = x 7 is D neither linear nor homogeneous Solution: Linearity is ruinied by the y 3 term; homogeneity is ruined by the x 7 on the
More informationdy x a. Sketch the slope field for the points: (1,±1), (2,±1), ( 1, ±1), and (0,±1).
Chapter 6. d x Given the differential equation: dx a. Sketch the slope field for the points: (,±), (,±), (, ±), and (0,±). b. Find the general solution for the given differential equation. c. Find the
More informationClosing Wed: HW_9A,9B (9.3/4,3.8) Final: Sat, June 3 th, 1:30-4:20, ARC 147
Closing Wed: HW_9A,9B (9.3/4,3.8) Final: Sat, June 3 th, 1:30-4:20, ARC 147 New material for the final, be able to: Solve separable diff. eq.. Use initial conditions & constants. Be able to set up the
More informationModeling using ODEs: Mixing Tank Problem. Natasha Sharma, Ph.D.
Reference W. Boyce and R. DiPrima, Elementary Differential Equations and Boundary Value s 8th Edition, John Wiley and Sons, 2005. First-Order Ordinary Differential Equation Definition (First Order Ordinary
More informationHOMEWORK # 3 SOLUTIONS
HOMEWORK # 3 SOLUTIONS TJ HITCHMAN. Exercises from the text.. Chapter 2.4. Problem 32 We are to use variation of parameters to find the general solution to y + 2 x y = 8x. The associated homogeneous equation
More information10 Exponential and Logarithmic Functions
10 Exponential and Logarithmic Functions Concepts: Rules of Exponents Exponential Functions Power Functions vs. Exponential Functions The Definition of an Exponential Function Graphing Exponential Functions
More informationSection Exponential Functions
121 Section 4.1 - Exponential Functions Exponential functions are extremely important in both economics and science. It allows us to discuss the growth of money in a money market account as well as the
More informationOrdinary Differential Equations
12/01/2015 Table of contents Second Order Differential Equations Higher Order Differential Equations Series The Laplace Transform System of First Order Linear Differential Equations Nonlinear Differential
More informationMATH 251 Examination I February 25, 2016 FORM A. Name: Student Number: Section:
MATH 251 Examination I February 25, 2016 FORM A Name: Student Number: Section: This exam has 13 questions for a total of 100 points. Show all your work! In order to obtain full credit for partial credit
More informationOn linear and non-linear equations.(sect. 2.4).
On linear and non-linear equations.sect. 2.4). Review: Linear differential equations. Non-linear differential equations. Properties of solutions to non-linear ODE. The Bernoulli equation. Review: Linear
More informationMath Applied Differential Equations
Math 256 - Applied Differential Equations Notes Basic Definitions and Concepts A differential equation is an equation that involves one or more of the derivatives (first derivative, second derivative,
More informationSPS Mathematical Methods Lecture #7 - Applications of First-order Differential Equations
1. Linear Models SPS 2281 - Mathematical Methods Lecture #7 - Applications of First-order Differential Equations (a) Growth and Decay (b) Half-life of Radioactive (c) Carbon Dating (d) Newton s Law of
More informationMATH 320, WEEK 4: Exact Differential Equations, Applications
MATH 320, WEEK 4: Exact Differential Equations, Applications 1 Exact Differential Equations We saw that the trick for first-order differential equations was to recognize the general property that the product
More informationMATH 251 Examination I October 5, 2017 FORM A. Name: Student Number: Section:
MATH 251 Examination I October 5, 2017 FORM A Name: Student Number: Section: This exam has 13 questions for a total of 100 points. Show all your work! In order to obtain full credit for partial credit
More informationNotes for exponential functions The week of March 6. Math 140
Notes for exponential functions The week of March 6 Math 140 Exponential functions: formulas An exponential function has the formula f (t) = ab t, where b is a positive constant; a is the initial value
More informationMath 266, Midterm Exam 1
Math 266, Midterm Exam 1 February 19th 2016 Name: Ground Rules: 1. Calculator is NOT allowed. 2. Show your work for every problem unless otherwise stated (partial credits are available). 3. You may use
More informationLinear Variable coefficient equations (Sect. 1.2) Review: Linear constant coefficient equations
Linear Variable coefficient equations (Sect. 1.2) Review: Linear constant coefficient equations. The Initial Value Problem. Linear variable coefficients equations. The Bernoulli equation: A nonlinear equation.
More informationChapters 8.1 & 8.2 Practice Problems
EXPECTED SKILLS: Chapters 8.1 & 8. Practice Problems Be able to verify that a given function is a solution to a differential equation. Given a description in words of how some quantity changes in time
More informationWrite each expression as a sum or difference of logarithms. All variables are positive. 4) log ( ) 843 6) Solve for x: 8 2x+3 = 467
Write each expression as a single logarithm: 10 Name Period 1) 2 log 6 - ½ log 9 + log 5 2) 4 ln 2 - ¾ ln 16 Write each expression as a sum or difference of logarithms. All variables are positive. 3) ln
More informationy = (1 y)cos t Answer: The equation is not autonomous because of the cos t term.
Math 211 Homework #4 Februar 9, 2001 2.9.2. = 1 2 + 2 Answer: Note that = 1 2 + 2 is autonomous, having form = f(). Solve the equation f()= 0 to find the equilibrium points. f()= 0 1 2 + 2 = 0 = 1. Thus,
More informationy0 = F (t0)+c implies C = y0 F (t0) Integral = area between curve and x-axis (where I.e., f(t)dt = F (b) F (a) wheref is any antiderivative 2.
Calulus pre-requisites you must know. Derivative = slope of tangent line = rate. Integral = area between curve and x-axis (where area can be negative). The Fundamental Theorem of Calculus: Suppose f continuous
More informationCalculus IV - HW 1. Section 20. Due 6/16
Calculus IV - HW Section 0 Due 6/6 Section.. Given both of the equations y = 4 y and y = 3y 3, draw a direction field for the differential equation. Based on the direction field, determine the behavior
More informationAssignment # 3, Math 370, Fall 2018 SOLUTIONS:
Assignment # 3, Math 370, Fall 2018 SOLUTIONS: Problem 1: Solve the equations: (a) y (1 + x)e x y 2 = xy, (i) y(0) = 1, (ii) y(0) = 0. On what intervals are the solution of the IVP defined? (b) 2y + y
More information2.4 Differences Between Linear and Nonlinear Equations 75
.4 Differences Between Linear and Nonlinear Equations 75 fying regions of the ty-plane where solutions exhibit interesting features that merit more detailed analytical or numerical investigation. Graphical
More informationMath M111: Lecture Notes For Chapter 10
Math M: Lecture Notes For Chapter 0 Sections 0.: Inverse Function Inverse function (interchange and y): Find the equation of the inverses for: y = + 5 ; y = + 4 3 Function (from section 3.5): (Vertical
More informationDIFFERENTIATION RULES
3 DIFFERENTIATION RULES DIFFERENTIATION RULES 3.8 Exponential Growth and Decay In this section, we will: Use differentiation to solve real-life problems involving exponentially growing quantities. EXPONENTIAL
More informationThe final is comprehensive (8-9 pages). There will be two pages on ch 9.
Closing Wed: HW_9A,9B (9.3/4,3.8) Final: Sat, Dec. 9 th, 1:30-4:20, KANE 130 Assigned seats, for your seat go to: catalyst.uw.edu/gradebook/aloveles/102715 The final is comprehensive (8-9 pages). There
More informationIntroduction to di erential equations
Chapter 1 Introduction to di erential equations 1.1 What is this course about? A di erential equation is an equation where the unknown quantity is a function, and where the equation involves the derivative(s)
More informationUCLA: Math 3B Problem set 7 (solutions) Fall, 2017
This week you will get practice solving separable differential equations, as well as some practice with linear models *Numbers in parentheses indicate the question has been taken from the textbook: S.
More informationModeling with First-Order Equations
Modeling with First-Order Equations MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Spring 2018 Radioactive Decay Radioactive decay takes place continuously. The number
More informationDrill Exercise Differential equations by abhijit kumar jha DRILL EXERCISE - 1 DRILL EXERCISE - 2. e x, where c 1
DRILL EXERCISE -. Find the order and degree (if defined) of the differential equation 5 4 3 d y d y. Find the order and degree (if defined) of the differential equation n. 3. Find the order and degree
More informationSummer 2017 Session 1 Math 2410Q (Section 10) Elementary Differential Equations M-Th 4:45pm-7:00pm
Summer 2017 Session 1 Math 2410Q (Section 10) Elementary Differential Equations M-Th 4:45pm-7:00pm Instructor: Dr. Angelynn Alvarez E-mail: angelynn.alvarez@uconn.edu Office: MONT 305 Office Hours: MTuTh
More informationLogarithmic and Exponential Equations and Inequalities College Costs
Logarithmic and Exponential Equations and Inequalities ACTIVITY 2.6 SUGGESTED LEARNING STRATEGIES: Summarize/ Paraphrase/Retell, Create Representations Wesley is researching college costs. He is considering
More information1. The accumulated net change function or area-so-far function
Name: Section: Names of collaborators: Main Points: 1. The accumulated net change function ( area-so-far function) 2. Connection to antiderivative functions: the Fundamental Theorem of Calculus 3. Evaluating
More informationMATH 2410 PRACTICE PROBLEMS FOR FINAL EXAM
MATH 2410 PRACTICE PROBLEMS FOR FINAL EXAM Date and place: Saturday, December 16, 2017. Section 001: 3:30-5:30 pm at MONT 225 Section 012: 8:00-10:00am at WSRH 112. Material covered: Lectures, quizzes,
More informationMath 3313: Differential Equations First-order ordinary differential equations
Math 3313: Differential Equations First-order ordinary differential equations Thomas W. Carr Department of Mathematics Southern Methodist University Dallas, TX Outline Math 2343: Introduction Separable
More informationChapter 2 Describing Change: Rates
Chapter Describing Change: Rates Section.1 Change, Percentage Change, and Average Rates of Change 1. 3. $.30 $0.46 per day 5 days = The stock price rose an average of 46 cents per day during the 5-day
More informationChapter 11 Logarithms
Chapter 11 Logarithms Lesson 1: Introduction to Logs Lesson 2: Graphs of Logs Lesson 3: The Natural Log Lesson 4: Log Laws Lesson 5: Equations of Logs using Log Laws Lesson 6: Exponential Equations using
More informationMATH 1231 MATHEMATICS 1B Calculus Section 3A: - First order ODEs.
MATH 1231 MATHEMATICS 1B 2010. For use in Dr Chris Tisdell s lectures. Calculus Section 3A: - First order ODEs. Created and compiled by Chris Tisdell S1: What is an ODE? S2: Motivation S3: Types and orders
More informationMathematics 134 Calculus 2 With Fundamentals Exam 4 Solutions for Review Sheet Sample Problems April 26, 2018
Mathematics 3 Calculus 2 With Fundamentals Exam Solutions for Review Sheet Sample Problems April 26, 28 Sample problems Note: The actual exam will be considerably shorter than the following list of questions
More informationMA 226 FINAL EXAM. Show Your Work. Problem Possible Actual Score
Name: MA 226 FINAL EXAM Show Your Work Problem Possible Actual Score 1 36 2 8 3 8 4 8 5 8 6 8 7 8 8 8 9 8 TOTAL 100 1.) 30 points (3 each) Short Answer: The answers to these questions need only consist
More informationSample Questions, Exam 1 Math 244 Spring 2007
Sample Questions, Exam Math 244 Spring 2007 Remember, on the exam you may use a calculator, but NOT one that can perform symbolic manipulation (remembering derivative and integral formulas are a part of
More informationReview Assignment II
MATH 11012 Intuitive Calculus KSU Name:. Review Assignment II 1. Let C(x) be the cost, in dollars, of manufacturing x widgets. Fill in the table with a mathematical expression and appropriate units corresponding
More informationIntroduction to Differential Equations Math 286 X1 Fall 2009 Homework 2 Solutions
Introuction to Differential Equations Math 286 X1 Fall 2009 Homewk 2 Solutions 1. Solve each of the following ifferential equations: (a) y + 3xy = 0 (b) y + 3y = 3x (c) y t = cos(t)y () x 2 y x y = 3 Solution:
More informationForm A. 1. Which of the following is a second-order, linear, homogenous differential equation? 2
Form A Math 4 Common Part of Final Exam December 6, 996 INSTRUCTIONS: Please enter your NAME, ID NUMBER, FORM designation, and INDEX NUMBER on your op scan sheet. The index number should be written in
More information( ) ( ). ( ) " d#. ( ) " cos (%) " d%
Math 22 Fall 2008 Solutions to Homework #6 Problems from Pages 404-407 (Section 76) 6 We will use the technique of Separation of Variables to solve the differential equation: dy d" = ey # sin 2 (") y #
More informationREVIEW PROBLEMS FOR MIDTERM I MATH 2373, SPRING 2019 UNIVERSITY OF MINNESOTA ANSWER KEY
REVIEW PROBLEMS FOR MIDTERM I MATH 2373, SPRING 209 UNIVERSITY OF MINNESOTA ANSWER KEY This list of problems is not guaranteed to be a complete review. For a complete review make sure that you know how
More informationMath 116 Practice for Exam 2
Math 6 Practice for Exam 2 Generated October 29, 205 Name: SOLUTIONS Instructor: Section Number:. This exam has 7 questions. Note that the problems are not of equal difficulty, so you may want to skip
More informationSolutions x. Figure 1: g(x) x g(t)dt ; x 0,
MATH Quiz 4 Spring 8 Solutions. (5 points) Express ln() in terms of ln() and ln(3). ln() = ln( 3) = ln( ) + ln(3) = ln() + ln(3). (5 points) If g(x) is pictured in Figure and..5..5 3 4 5 6 x Figure : g(x)
More informationThe units on the average rate of change in this situation are. change, and we would expect the graph to be. ab where a 0 and b 0.
Lesson 9: Exponential Functions Outline Objectives: I can analyze and interpret the behavior of exponential functions. I can solve exponential equations analytically and graphically. I can determine the
More information