# Solutions to Math 53 First Exam April 20, 2010

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 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. (The scale on each is x, y.) I II III IV V VI Equation I, II, III, IV, V VI, or none Equation I, II, III, IV, V VI, or none y = y II y = e y IV y = y + VI y = x + none y = y I y = x III y = x y V y = y none

2 Math 53, Spring 00 Solutions to First Exam April 0, 00 Page of 9. (0 points) (a) Give an interval in which the initial value problem (4 t )y + ty = 3t, y( 3) = has a unique solution; give your reasoning. (Note: you are not being asked to solve the differential equation!) (0 points) There are two plausible solutions. In the first place we can write the ode as y + t 4 t y = 3t 4 t which is a linear first order ode. We can now cite Theorem.4. of the book, which guarantees the existence (and uniqueness) of a solution y = φ(t), defined on an open interval containing t 0 = 3 where the functions are continuous. But these rational functions of t are t 4 t and 3t 4 t continuous so long as t ±, so the open interval we seek is (, ). Alternatively, we can rewrite the ode in the form y = 3t 4 t ty 4 t With this formulation, one can write y = F (t, y), and then apply Theorem.4. of the book on existence & uniqueness of solutions to first-order ivps. It suffices to check that F (t, y) and F y (t, y) are continuous in a neighbourhood of (t 0, y 0 ) = ( 3, ), which is easily done since F (t, y) and F y (t, y) are defined and continuous provided t ±. The result then states that there exists a solution y = φ(t) on some interval ( 3 h, 3 + h). This result, importantly, does not guarantee the existence of a solution y = φ(t) for all t (, ).

3 Math 53, Spring 00 Solutions to First Exam April 0, 00 Page 3 of 9 (b) Construct a first-order linear differential equation whose solutions all approach the curve y = t as t. Then solve your ODE and show that the solutions have this property. (0 points) One way to construct an ode with the required property is to obtain first an ode all of whose solutions, Y = φ(t), have the property that lim t φ(t) = 0. One such is given by Y + Y = 0 which has general solution Y = ce t. We then write y c (t) = ce t + ( t ), and observe that y c (t) satisfies the ode y + y = ce t + ( t ) + ce t + ( t ) = t + t We have obtained a candidate ode: y + y = t t. There are several ways to solve this in full, using integrating factors or undetermined coefficients. We use the laziest method. Suppose φ(t) solves the ode; then since the ode is linear and the coefficient functions are continuous at all t, the domain of definition of φ can be assumed to be (, ). In particular, φ(0) = y 0 is defined. Consider the function ψ(t) = ( y 0 )e t + ( t ) We have ψ(0) = y 0 + = φ(0), and φ, ψ both satisfy the constructed linear first order ode. By the uniqueness part of the existence and uniqueness theorem for first order linear odes, they must coincide. All solutions of the ode therefore being of the form ce t + t, they all have the required property.

4 Math 53, Spring 00 Solutions to First Exam April 0, 00 Page 4 of 9 3. (5 points) A 000 gallon pool initially contains 500 gallons of water with 500 milligrams of a chemical mixed in. At noon, a solution containing 4 milligrams of chemical per gallon of water begins entering at a rate of gallons per minute, while the well-mixed liquid flows out of the pool at a rate of gallons per minute. Let C(t) denote the amount (in milligrams) of the chemical in the pool at t minutes after noon. (a) Write a differential equation satisfied by C(t), and give the value of C(0). (4 points) At time t, the rate at which chemical is entering the pool is 4 mg/gal gal/min = 8 mg/min, and the rate at which chemical is leaving the pool is gal/min C(t) mg 500 gal = C(t) 50 mg/min. Thus C satisfies the differential equation C = 8 C 50 (mg/min). The initial value is C(0) = 500 mg. (b) Solve for C(t), showing all steps; your answer should not depend on any unknown constants. (6 points) Separating variables, we get the equation dc 8 C/50 = dt, which is equivalent to dc 000 C = dt 50. Integrating on both sides, we get the equation which gives C 500 ds t 000 s = ds 0 50, ln 000 C 500 = t 50. Multiplying by ( ) and taking the exponential function on both sides, we get 000 C 500 = e t/50. It follows that 000 C 500 = ±e t/50, which gives C = 000 ± 500e t/50. Only the solution with the minus sign satisfies the initial condition C(0) = 500, so we have C(t) = e t/50.

5 Math 53, Spring 00 Solutions to First Exam April 0, 00 Page 5 of 9 (c) What happens to C(t) as t? Explain. ( points) When t, the term 500e t/50 0, and hence C(t) = e t/ Thus the amount of chemical in the pool approaches 000 mg as t. (d) Suppose that the situation is exactly as described above, except that the inflow rate is 3 gallons per minute (and the outflow rate remains at gallons per minute). Write a differential equation satisfied by C(t) in this situation. (Assume 0 t 500.) You do not need to solve this differential equation. (3 points) The amount of water in the pool at time t is 500 gal + t min (3 ) gal/min = (500 + t) gal. At time t, the rate at which chemical is entering the pool is 4 mg/gal 3 gal/min = mg/min. and the rate at which chemical is leaving the pool is Thus C satisfies the equation gal/min C(t) mg (500 + t) gal = C(t) t mg/min. C = C t (mg/min).

6 Math 53, Spring 00 Solutions to First Exam April 0, 00 Page 6 of 9 4. (0 points) Find a solution y(t) to the initial value problem yy + t = 0, y(0) = 3 and specify the solution s interval of definition (i.e. domain of definition). The equation is separable; we may rearrange both sides and integrate to find y dy = t dt so that y = C t for some C. (Alternatively, our original ODE can be rewritten as d dt (y + t ) = 0, so that as a function of t, y + t is a constant.) We can evaluate C by plugging in the initial condition y(0) = 3; we find C = 9/. Thus Solving for y, we obtain y = 9 t. y = ± 9 t The solution has two branches; we must pick the one with minus sign because y(0) = 3. Hence y = 9 t The interval of definition is ( 3, 3). Notice the endpoints ±3 do not belong to the interval, because y(t) is not differentiable at these points.

7 Math 53, Spring 00 Solutions to First Exam April 0, 00 Page 7 of 9 5. (0 points) Find a solution to the differential equation satisfying y() =. dy dx = yexy x xe xy We can re-write the equation as Let M = ye xy + x and N = xe xy. Then (ye xy + x) + (xe xy ) dy dx = 0. M y = xye xy + e xy and N x = xye xy + e xy are equal, and so the equation is exact. We wish to find a function ψ(x, y) such that ψ x = M and ψ y = N. Integrating M with respect to x we find that ψ(x, y) = Mdx = exy + x + h(y). To find h(y) we use the condition ψ y = N: ψ y = xe xy + h (y) = N = xe xy to conclude that h (y) = 0, and so h(y) = c is a constant. Therefore ψ(x, y) = exy + x + c. Any solution y(x) satisfies ψ(x, y(x)) = c for some constant c. Therefore, or, setting c = (c c ), this is equivalent to exy + x + c = c e xy + x = c. The initial condition y() = is obeyed if e + = c. Therefore the solution can be written implicitly as e xy + x = e +. But this can be solved for y: e xy = e + x xy = ln(e + x ) y = x ln(e + x ).

8 Math 53, Spring 00 Solutions to First Exam April 0, 00 Page 8 of 9 6. (0 points) Consider the autonomous differential equation dy dt = y 9y/3. (a) One equilibrium solution is y = 0. Now determine all other equilibria, and for each nonzero equilibrium, classify it as stable or unstable; give reasoning. (You do not need to discuss the status of y = 0.) (4 points) Let f(y) = y 9y /3. We first notice that the autonomous ODE can be written dy dt = f(y) = y 9y/3 = y /3 (y /3 9) = y /3 (y /3 3)(y /3 + 3) and that the function f is continuous everywhere, but that f y = f (y) = 3y /3 fails to be continuous exactly on the line {y = 0}. As a consequence, away from this line the equilibria are exactly those values y so that f(y) = 0. By the above factorization, this gives the non-zero equilibria y = 7 and y = 7 (note that we get both signs). Each of these equilibria is unstable, since f is positive there (see also footnote, next page): f (7) = 3(7) /3 = 3 > 0, f ( 7) = 3( 7) /3 = 3 > 0. (b) Determine precisely the regions where the graph of a solution curve (y versus t) is concave up, and where it is concave down. Show your reasoning. (4 points) We use the chain rule to compute: d y dt = d dt f(y) = f (y) dy dt = f (y)f(y) = ( 3y /3 )(y 9y /3 ) = y /3 (y /3 3)(y /3 9). Thus, potential sign changes take place at y = ±3 3/ = ± 7 = ±3 3, and also at the three equilibria y = 0, ±7. These give six intervals of y-values to check the product s sign; we find d y dt > 0 y = y(t) concave up for: 7 < y < 7, 0 < y < 7, y > 7; and d y dt < 0 y = y(t) concave down for: y < 7, 7 < y < 0, 7 < y < 7. (c) On the axes below, sketch several solution curves (y versus t) satisfying y(0) = y 0 for various values of y 0. You must sketch enough curves to clearly depict all of the different behaviors you determined in parts (a) and (b), and label your axis to include the key values you found in (a) and (b). (But: you do not need to find explicit solutions.) (4 points) In addition to the equilibria, at least four curves (one in each region formed by the three equilibria) should be drawn, and the location of inflection points (i.e., concavity change) clearly labeled. As an example, the curves depicted at right roughly correspond to initial values y 0 = 30, 5,,, 30, as well as the equilibrium values 0, ±7.

9 Math 53, Spring 00 Solutions to First Exam April 0, 00 Page 9 of 9 Footnote to (a): Recall why f (y 0 ) > 0 implies the equilibrium y = y 0 is unstable: this says that for y slightly less than y 0, then f(y) is negative, and so y is decreasing and thus moving down, i.e. away from the equilibrium. On the other hand, when y is slightly above the equilibrium y 0, then f(y) is positive, and so y is increasing and thus moving up, i.e. away from the equilibrium. (d) Find explicit expressions for three distinct functions y(t), each defined for all real values of t, and each satisfying the following initial value problem: dy dt = y 9y/3 y(3) = 0. for all t in (, ), (4 points) We note that y (t) = 0 is readily seen to satisfy both the ODE and the initial conditon and so is one possible solution. In order to get other solutions, we could separate variables and integrate (since any autonomous equation is separable). The integration, seemingly difficult, is greatly simplified via the substitution u = y /3. (Try this!) Alternatively, we could simply transform the original ODE using this change of variables u = y /3. We then have that u satisfies (by the chain rule): du dt = dy y /3 3 dt = 3 y /3 (y 9y /3 ) = (u 9). 3 It then follows that u(t) = 9 + Ce /3t. As a consequnce we have y(t) = ± (9 + Ce t/3) 3/ In order to have y(3) = 0, we must take C = 9e. But this gives a function defined only on t 3, since the half-power fails to be defined when 9 + Ce t/3 < 0. However, it is not hard to check that the piecewise function: { (9 9e t/3 ) 3/ t < 3 y (t) = 0 t 3 is continuous and differentiable for all t and satisfies both the equation and the initial condition. The same can be said of the function y 3 (t) = y (t), which is clearly distinct from y (t). Thus y (t), y (t), y 3 (t) are three distinct solutions. (e) Discuss in precise terms how the phenomenon of part (d) can be reconciled with the existence and uniqueness theorem for nonlinear ODEs. (You can answer this question without having found a complete answer to part (d).) (4 points) As mentioned before, the function f(y) fails to have a continuous derivative at y = 0. This means that the existence and uniqueness theorem tells us nothing about the uniqueness of solutions that pass through points of the form (t, 0). In particular, there may be be multiple solutions to the ODE through a given point of (t, 0). Indeed, we give three such solutions through (3, 0).

### Homework 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

### Solutions of Math 53 Midterm Exam I

Solutions of Math 53 Midterm Exam I Problem 1: (1) [8 points] Draw a direction field for the given differential equation y 0 = t + y. (2) [8 points] Based on the direction field, determine the behavior

### First 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

### Math 2a Prac Lectures on Differential Equations

Math 2a Prac Lectures on Differential Equations Prof. Dinakar Ramakrishnan 272 Sloan, 253-37 Caltech Office Hours: Fridays 4 5 PM Based on notes taken in class by Stephanie Laga, with a few added comments

### HW2 Solutions. MATH 20D Fall 2013 Prof: Sun Hui TA: Zezhou Zhang (David) October 14, Checklist: Section 2.6: 1, 3, 6, 8, 10, 15, [20, 22]

HW2 Solutions MATH 20D Fall 2013 Prof: Sun Hui TA: Zezhou Zhang (David) October 14, 2013 Checklist: Section 2.6: 1, 3, 6, 8, 10, 15, [20, 22] Section 3.1: 1, 2, 3, 9, 16, 18, 20, 23 Section 3.2: 1, 2,

### MATH 307 Introduction to Differential Equations Autumn 2017 Midterm Exam Monday November

MATH 307 Introduction to Differential Equations Autumn 2017 Midterm Exam Monday November 6 2017 Name: Student ID Number: I understand it is against the rules to cheat or engage in other academic misconduct

### First Order ODEs, Part I

Craig J. Sutton craig.j.sutton@dartmouth.edu Department of Mathematics Dartmouth College Math 23 Differential Equations Winter 2013 Outline 1 2 in General 3 The Definition & Technique Example Test for

### (1 + 2y)y = x. ( x. The right-hand side is a standard integral, so in the end we have the implicit solution. y(x) + y 2 (x) = x2 2 +C.

Midterm 1 33B-1 015 October 1 Find the exact solution of the initial value problem. Indicate the interval of existence. y = x, y( 1) = 0. 1 + y Solution. We observe that the equation is separable, and

### Problem 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

### Do not write below here. Question Score Question Score Question Score

MATH-2240 Friday, May 4, 2012, FINAL EXAMINATION 8:00AM-12:00NOON Your Instructor: Your Name: 1. Do not open this exam until you are told to do so. 2. This exam has 30 problems and 18 pages including this

### 20D - Homework Assignment 1

0D - Homework Assignment Brian Bowers (TA for Hui Sun) MATH 0D Homework Assignment October 7, 0. #,,,4,6 Solve the given differential equation. () y = x /y () y = x /y( + x ) () y + y sin x = 0 (4) y =

### Calculus I Review Solutions

Calculus I Review Solutions. Compare and contrast the three Value Theorems of the course. When you would typically use each. The three value theorems are the Intermediate, Mean and Extreme value theorems.

### (1) Rate of change: A swimming pool is emptying at a constant rate of 90 gal/min.

CHAPTER 1 Introduction 1. Bacground Models of physical situations from Calculus (1) Rate of change: A swimming pool is emptying at a constant rate of 90 gal/min. With V = volume in gallons and t = time

### ANSWERS Final Exam Math 250b, Section 2 (Professor J. M. Cushing), 15 May 2008 PART 1

ANSWERS Final Exam Math 50b, Section (Professor J. M. Cushing), 5 May 008 PART. (0 points) A bacterial population x grows exponentially according to the equation x 0 = rx, where r>0is the per unit rate

### Practice Midterm Solutions

Practice Midterm Solutions Math 4B: Ordinary Differential Equations Winter 20 University of California, Santa Barbara TA: Victoria Kala DO NOT LOOK AT THESE SOLUTIONS UNTIL YOU HAVE ATTEMPTED EVERY PROBLEM

### Applied Calculus I Practice Final Exam Solution Notes

AMS 5 (Fall, 2009). Solve for x: 0 3 2x = 3 (.2) x Taking the natural log of both sides, we get Applied Calculus I Practice Final Exam Solution Notes Joe Mitchell ln 0 + 2xln 3 = ln 3 + xln.2 x(2ln 3 ln.2)

### Solutions 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

### dt 2 The Order of a differential equation is the order of the highest derivative that occurs in the equation. Example The differential equation

Lecture 18 : Direction Fields and Euler s Method A Differential Equation is an equation relating an unknown function and one or more of its derivatives. Examples Population growth : dp dp = kp, or = kp

### Modeling with differential equations

Mathematical Modeling Lia Vas Modeling with differential equations When trying to predict the future value, one follows the following basic idea. Future value = present value + change. From this idea,

### Definition of differential equations and their classification. Methods of solution of first-order differential equations

Introduction to differential equations: overview Definition of differential equations and their classification Solutions of differential equations Initial value problems Existence and uniqueness Mathematical

### Review Sheet 2 Solutions

Review Sheet Solutions. A bacteria culture initially contains 00 cells and grows at a rate proportional to its size. After an hour the population has increased to 40 cells. (a) Find an expression for the

### MATH 319, WEEK 2: Initial Value Problems, Existence/Uniqueness, First-Order Linear DEs

MATH 319, WEEK 2: Initial Value Problems, Existence/Uniqueness, First-Order Linear DEs 1 Initial-Value Problems We have seen that differential equations can, in general, given rise to multiple solutions.

### Differential 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

### Solutions to Final Exam Sample Problems, Math 246, Spring 2011

Solutions to Final Exam Sample Problems, Math 246, Spring 2 () Consider the differential equation dy dt = (9 y2 )y 2 (a) Identify its equilibrium (stationary) points and classify their stability (b) Sketch

### MATH 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

### Math 116 Second Midterm March 20, 2017

EXAM SOLUTIONS Math 6 Second Midterm March 0, 07. Do not open this exam until you are told to do so.. Do not write your name anywhere on this exam. 3. This exam has pages including this cover. There are

### Solving Differential Equations: First Steps

30 ORDINARY DIFFERENTIAL EQUATIONS 3 Solving Differential Equations Solving Differential Equations: First Steps Now we start answering the question which is the theme of this book given a differential

### Chapter 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

### Introduction 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)

### 4 Partial Differentiation

4 Partial Differentiation Many equations in engineering, physics and mathematics tie together more than two variables. For example Ohm s Law (V = IR) and the equation for an ideal gas, PV = nrt, which

### 1+t 2 (l) y = 2xy 3 (m) x = 2tx + 1 (n) x = 2tx + t (o) y = 1 + y (p) y = ty (q) y =

DIFFERENTIAL EQUATIONS. Solved exercises.. Find the set of all solutions of the following first order differential equations: (a) x = t (b) y = xy (c) x = x (d) x = (e) x = t (f) x = x t (g) x = x log

### MAT 311 Midterm #1 Show your work! 1. The existence and uniqueness theorem says that, given a point (x 0, y 0 ) the ODE. y = (1 x 2 y 2 ) 1/3

MAT 3 Midterm # Show your work!. The existence and uniqueness theorem says that, given a point (x 0, y 0 ) the ODE y = ( x 2 y 2 ) /3 has a unique (local) solution with initial condition y(x 0 ) = y 0

### Series Solutions. 8.1 Taylor Polynomials

8 Series Solutions 8.1 Taylor Polynomials Polynomial functions, as we have seen, are well behaved. They are continuous everywhere, and have continuous derivatives of all orders everywhere. It also turns

### MA26600 FINAL EXAM INSTRUCTIONS December 13, You must use a #2 pencil on the mark sense sheet (answer sheet).

MA266 FINAL EXAM INSTRUCTIONS December 3, 2 NAME INSTRUCTOR. You must use a #2 pencil on the mark sense sheet (answer sheet). 2. On the mark-sense sheet, fill in the instructor s name (if you do not know,

### Solutions to Exam 1, Math Solution. Because f(x) is one-to-one, we know the inverse function exists. Recall that (f 1 ) (a) =

Solutions to Exam, Math 56 The function f(x) e x + x 3 + x is one-to-one (there is no need to check this) What is (f ) ( + e )? Solution Because f(x) is one-to-one, we know the inverse function exists

### FIRST-ORDER SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS III: Autonomous Planar Systems David Levermore Department of Mathematics University of Maryland

FIRST-ORDER SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS III: Autonomous Planar Systems David Levermore Department of Mathematics University of Maryland 4 May 2012 Because the presentation of this material

### Lecture 10. (2) Functions of two variables. Partial derivatives. Dan Nichols February 27, 2018

Lecture 10 Partial derivatives Dan Nichols nichols@math.umass.edu MATH 233, Spring 2018 University of Massachusetts February 27, 2018 Last time: functions of two variables f(x, y) x and y are the independent

### Study Guide/Practice Exam 3

Study Guide/Practice Exam 3 This study guide/practice exam covers only the material since exam. The final exam, however, is cumulative so you should be sure to thoroughly study earlier material. The distribution

### Chapter 4. Section Derivatives of Exponential and Logarithmic Functions

Chapter 4 Section 4.2 - Derivatives of Exponential and Logarithmic Functions Objectives: The student will be able to calculate the derivative of e x and of lnx. The student will be able to compute the

### Math 116 Second Midterm November 14, 2012

Math 6 Second Midterm November 4, Name: EXAM SOLUTIONS Instructor: Section:. Do not open this exam until you are told to do so.. This exam has pages including this cover. There are 8 problems. Note that

### Section , #5. Let Q be the amount of salt in oz in the tank. The scenario can be modeled by a differential equation.

Section.3.5.3, #5. Let Q be the amount of salt in oz in the tank. The scenario can be modeled by a differential equation dq = 1 4 (1 + sin(t) ) + Q, Q(0) = 50. (1) 100 (a) The differential equation given

### 3.9 Derivatives of Exponential and Logarithmic Functions

322 Chapter 3 Derivatives 3.9 Derivatives of Exponential and Logarithmic Functions Learning Objectives 3.9.1 Find the derivative of exponential functions. 3.9.2 Find the derivative of logarithmic functions.

### Section 1.4 Circles. Objective #1: Writing the Equation of a Circle in Standard Form.

1 Section 1. Circles Objective #1: Writing the Equation of a Circle in Standard Form. We begin by giving a definition of a circle: Definition: A Circle is the set of all points that are equidistant from

### Polytechnic Institute of NYU MA 2132 Final Practice Answers Fall 2012

Polytechnic Institute of NYU MA Final Practice Answers Fall Studying from past or sample exams is NOT recommended. If you do, it should be only AFTER you know how to do all of the homework and worksheet

### Mathematic 108, Fall 2015: Solutions to assignment #7

Mathematic 08, Fall 05: Solutions to assignment #7 Problem # Suppose f is a function with f continuous on the open interval I and so that f has a local maximum at both x = a and x = b for a, b I with a

### On linear and non-linear equations. (Sect. 1.6).

On linear and non-linear equations. (Sect. 1.6). Review: Linear differential equations. Non-linear differential equations. The Picard-Lindelöf Theorem. Properties of solutions to non-linear ODE. The Proof

### AMATH 351 Mar 15, 2013 FINAL REVIEW. Instructor: Jiri Najemnik

AMATH 351 Mar 15, 013 FINAL REVIEW Instructor: Jiri Najemni ABOUT GRADES Scores I have so far will be posted on the website today sorted by the student number HW4 & Exam will be added early next wee Let

### 24. Partial Differentiation

24. Partial Differentiation The derivative of a single variable function, d f(x), always assumes that the independent variable dx is increasing in the usual manner. Visually, the derivative s value at

### MATH 108 FALL 2013 FINAL EXAM REVIEW

MATH 08 FALL 203 FINAL EXAM REVIEW Definitions and theorems. The following definitions and theorems are fair game for you to have to state on the exam. Definitions: Limit (precise δ-ɛ version; 2.4, Def.

### ORDINARY DIFFERENTIAL EQUATIONS

ORDINARY DIFFERENTIAL EQUATIONS GABRIEL NAGY Mathematics Department, Michigan State University, East Lansing, MI, 4884 NOVEMBER 9, 7 Summary This is an introduction to ordinary differential equations We

### Review for the Final Exam

Math 171 Review for the Final Exam 1 Find the limits (4 points each) (a) lim 4x 2 3; x x (b) lim ( x 2 x x 1 )x ; (c) lim( 1 1 ); x 1 ln x x 1 sin (x 2) (d) lim x 2 x 2 4 Solutions (a) The limit lim 4x

### Basic Theory of Differential Equations

page 104 104 CHAPTER 1 First-Order Differential Equations 16. The following initial-value problem arises in the analysis of a cable suspended between two fixed points y = 1 a 1 + (y ) 2, y(0) = a, y (0)

### Math 108, Solution of Midterm Exam 3

Math 108, Solution of Midterm Exam 3 1 Find an equation of the tangent line to the curve x 3 +y 3 = xy at the point (1,1). Solution. Differentiating both sides of the given equation with respect to x,

### Math 342 Partial Differential Equations «Viktor Grigoryan

Math 342 Partial Differential Equations «Viktor Grigoryan 15 Heat with a source So far we considered homogeneous wave and heat equations and the associated initial value problems on the whole line, as

### On 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

### MATH 251 Examination I July 5, 2011 FORM A. Name: Student Number: Section:

MATH 251 Examination I July 5, 2011 FORM A Name: Student Number: Section: This exam has 12 questions for a total of 100 points. Show all you your work! In order to obtain full credit for partial credit

### Properties of Derivatives

6 CHAPTER Properties of Derivatives To investigate derivatives using first principles, we will look at the slope of f ( ) = at the point P (,9 ). Let Q1, Q, Q, Q4, be a sequence of points on the curve

### Calculus IV - HW 3. Due 7/ Give the general solution to the following differential equations: y = c 1 e 5t + c 2 e 5t. y = c 1 e 2t + c 2 e 4t.

Calculus IV - HW 3 Due 7/13 Section 3.1 1. Give the general solution to the following differential equations: a y 25y = 0 Solution: The characteristic equation is r 2 25 = r 5r + 5. It follows that the

### Euler-Cauchy Using Undetermined Coefficients

Euler-Cauchy Using Undetermined Coefficients Department of Mathematics California State University, Fresno doreendl@csufresno.edu Joint Mathematics Meetings January 14, 2010 Outline 1 2 3 Second Order

### It 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

### SOLUTIONS Section (a) y' + 4 x+2 y = -6. (x+2) 2, P(x) = 4, P(x) dx = 4 applen(x+2) = applen(x+2)4. = (x+2) 4,

page of Section 4. solutions SOLUTIONS Section 4.. (a) y' + 4 x+2 y = -6 (x+2) 2, P(x) = 4, P(x) dx = 4 applen(x+2) = applen(x+2)4 x+2 applen(x+2) 4 I = e = (x+2) 4, answer is y = -2 x+2 + k (x+2) 4 (x+2)

### First and Second Order Differential Equations Lecture 4

First and Second Order Differential Equations Lecture 4 Dibyajyoti Deb 4.1. Outline of Lecture The Existence and the Uniqueness Theorem Homogeneous Equations with Constant Coefficients 4.2. The Existence

### Solutions to Math 41 Final Exam December 10, 2012

Solutions to Math 4 Final Exam December,. ( points) Find each of the following limits, with justification. If there is an infinite limit, then explain whether it is or. x ln(t + ) dt (a) lim x x (5 points)

### MATH 251 Examination I July 1, 2013 FORM A. Name: Student Number: Section:

MATH 251 Examination I July 1, 2013 FORM A Name: Student Number: Section: This exam has 12 questions for a total of 100 points. Show all your work! In order to obtain full credit for partial credit problems,

### z x = f x (x, y, a, b), z y = f y (x, y, a, b). F(x, y, z, z x, z y ) = 0. This is a PDE for the unknown function of two independent variables.

Chapter 2 First order PDE 2.1 How and Why First order PDE appear? 2.1.1 Physical origins Conservation laws form one of the two fundamental parts of any mathematical model of Continuum Mechanics. These

### MATH 1241 Common Final Exam Fall 2010

MATH 1241 Common Final Exam Fall 2010 Please print the following information: Name: Instructor: Student ID: Section/Time: The MATH 1241 Final Exam consists of three parts. You have three hours for the

### Math 240 Calculus III

Calculus III Summer 2015, Session II Monday, August 3, 2015 Agenda 1. 2. Introduction The reduction of technique, which applies to second- linear differential equations, allows us to go beyond equations

### AP Calculus Testbank (Chapter 6) (Mr. Surowski)

AP Calculus Testbank (Chapter 6) (Mr. Surowski) Part I. Multiple-Choice Questions 1. Suppose that f is an odd differentiable function. Then (A) f(1); (B) f (1) (C) f(1) f( 1) (D) 0 (E). 1 1 xf (x) =. The

### AP Calculus AB Winter Break Packet Happy Holidays!

AP Calculus AB Winter Break Packet 04 Happy Holidays! Section I NO CALCULATORS MAY BE USED IN THIS PART OF THE EXAMINATION. Directions: Solve each of the following problems. After examining the form of

### Math 216 First Midterm 19 October, 2017

Math 6 First Midterm 9 October, 7 This sample exam is provided to serve as one component of your studying for this exam in this course. Please note that it is not guaranteed to cover the material that

### Math 311, Partial Differential Equations, Winter 2015, Midterm

Score: Name: Math 3, Partial Differential Equations, Winter 205, Midterm Instructions. Write all solutions in the space provided, and use the back pages if you have to. 2. The test is out of 60. There

### MA 266 FINAL EXAM INSTRUCTIONS May 8, 2010

MA 266 FINAL EXAM INSTRUCTIONS May 8, 200 NAME INSTRUCTOR. You must use a #2 pencil on the mark sense sheet (answer sheet). 2. On the mark-sense sheet, fill in the instructor s name (if you do not know,

### Math 220 Final Exam Sample Problems December 12, Topics for Math Fall 2002

Math 220 Final Exam Sample Problems December 12, 2002 Topics for Math 220 - Fall 2002 Chapter 1. Solutions and Initial Values Approximation via the Euler method Chapter 2. First Order: Linear First Order:

### LAURENTIAN UNIVERSITY UNIVERSITÉ LAURENTIENNE

Page 1 of 15 LAURENTIAN UNIVERSITY UNIVERSITÉ LAURENTIENNE Wednesday, December 13 th Course and No. 2006 MATH 2066 EL Date................................... Cours et no........................... Total

### First and Second Order ODEs

Civil Engineering 2 Mathematics Autumn 211 M. Ottobre First and Second Order ODEs Warning: all the handouts that I will provide during the course are in no way exhaustive, they are just short recaps. Notation

### MATH 312 Section 8.3: Non-homogeneous Systems

MATH 32 Section 8.3: Non-homogeneous Systems Prof. Jonathan Duncan Walla Walla College Spring Quarter, 2007 Outline Undetermined Coefficients 2 Variation of Parameter 3 Conclusions Undetermined Coefficients

### Math 217 Fall 2000 Exam Suppose that y( x ) is a solution to the differential equation

Math 17 Fall 000 Exam 1 Notational Remark: In this exam, the symbol x y( x ) means dy dx. 1. Suppose that y( x ) is a solution to the differential equation, x y( x ) F ( x, y( x )) y( x ) 0 y 0. Then y'(x

### Even-Numbered Homework Solutions

Even-Numbered Homework Solutions Chapter 1 1.1 8. Using the decay-rate parameter you computed in 1.1.7, determine the time since death if: (a) 88% of the original C-14 is still in the material The decay-rate

### 1. Why don t we have to worry about absolute values in the general form for first order differential equations with constant coefficients?

1. Why don t we have to worry about absolute values in the general form for first order differential equations with constant coefficients? Let y = ay b with y(0) = y 0 We can solve this as follows y =

### THE USE OF A CALCULATOR, CELL PHONE, OR ANY OTHER ELECTRONIC DEVICE IS NOT PERMITTED IN THIS EXAMINATION.

MATH 110 FINAL EXAM SPRING 2008 FORM A NAME STUDENT NUMBER INSTRUCTOR SECTION NUMBER This examination will be machine processed by the University Testing Service. Use only a number 2 pencil on your scantron.

### Form 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

### M340 HW 2 SOLUTIONS. 1. For the equation y = f(y), where f(y) is given in the following plot:

M340 HW SOLUTIONS 1. For the equation y = f(y), where f(y) is given in the following plot: (a) What are the critical points? (b) Are they stable or unstable? (c) Sketch the solutions in the ty plane. (d)

### Exam Basics. midterm. 1 There will be 9 questions. 2 The first 3 are on pre-midterm material. 3 The next 1 is a mix of old and new material.

Exam Basics 1 There will be 9 questions. 2 The first 3 are on pre-midterm material. 3 The next 1 is a mix of old and new material. 4 The last 5 questions will be on new material since the midterm. 5 60

### MA 266 Review Topics - Exam # 2 (updated)

MA 66 Reiew Topics - Exam # updated Spring First Order Differential Equations Separable, st Order Linear, Homogeneous, Exact Second Order Linear Homogeneous with Equations Constant Coefficients The differential

### Math 2250 Final Exam Practice Problem Solutions. f(x) = ln x x. 1 x. lim. lim. x x = lim. = lim 2

Math 5 Final Eam Practice Problem Solutions. What are the domain and range of the function f() = ln? Answer: is only defined for, and ln is only defined for >. Hence, the domain of the function is >. Notice

### Matlab Problem Sets. Math 246 Spring 2012 Sections 0112, 0122, 0132, R. Lipsman

Matlab Problem Sets 1 Math 246 Spring 2012 Sections 0112, 0122, 0132, 0142 R. Lipsman 2 Problem Set A Practice with MATLAB In this problem set, you will use MATLAB to do some basic calculations, and then

### Least Squares Regression

Least Squares Regression Chemical Engineering 2450 - Numerical Methods Given N data points x i, y i, i 1 N, and a function that we wish to fit to these data points, fx, we define S as the sum of the squared

### DIFFERENTIAL EQUATIONS

DIFFERENTIAL EQUATIONS Basic Concepts Paul Dawkins Table of Contents Preface... Basic Concepts... 1 Introduction... 1 Definitions... Direction Fields... 8 Final Thoughts...19 007 Paul Dawkins i http://tutorial.math.lamar.edu/terms.aspx

### Graphical Relationships Among f, f,

Graphical Relationships Among f, f, and f The relationship between the graph of a function and its first and second derivatives frequently appears on the AP exams. It will appear on both multiple choice

### Applied Math Qualifying Exam 11 October Instructions: Work 2 out of 3 problems in each of the 3 parts for a total of 6 problems.

Printed Name: Signature: Applied Math Qualifying Exam 11 October 2014 Instructions: Work 2 out of 3 problems in each of the 3 parts for a total of 6 problems. 2 Part 1 (1) Let Ω be an open subset of R

### Math 308 Week 8 Solutions

Math 38 Week 8 Solutions There is a solution manual to Chapter 4 online: www.pearsoncustom.com/tamu math/. This online solutions manual contains solutions to some of the suggested problems. Here are solutions

### Math 256: Applied Differential Equations: Final Review

Math 256: Applied Differential Equations: Final Review Chapter 1: Introduction, Sec 1.1, 1.2, 1.3 (a) Differential Equation, Mathematical Model (b) Direction (Slope) Field, Equilibrium Solution (c) Rate

### Learning Target: I can sketch the graphs of rational functions without a calculator. a. Determine the equation(s) of the asymptotes.

Learning Target: I can sketch the graphs of rational functions without a calculator Consider the graph of y= f(x), where f(x) = 3x 3 (x+2) 2 a. Determine the equation(s) of the asymptotes. b. Find the

### Answer Key. Calculus I Math 141 Fall 2003 Professor Ben Richert. Exam 2

Answer Key Calculus I Math 141 Fall 2003 Professor Ben Richert Exam 2 November 18, 2003 Please do all your work in this booklet and show all the steps. Calculators and note-cards are not allowed. Problem

### LAURENTIAN UNIVERSITY UNIVERSITÉ LAURENTIENNE

Page 1 of 15 LAURENTIAN UNIVERSITY UNIVERSITÉ LAURENTIENNE Friday, December 14 th Course and No. 2007 MATH 2066 EL Date................................... Cours et no........................... Total no.

### Answer Key 1973 BC 1969 BC 24. A 14. A 24. C 25. A 26. C 27. C 28. D 29. C 30. D 31. C 13. C 12. D 12. E 3. A 32. B 27. E 34. C 14. D 25. B 26.

Answer Key 969 BC 97 BC. C. E. B. D 5. E 6. B 7. D 8. C 9. D. A. B. E. C. D 5. B 6. B 7. B 8. E 9. C. A. B. E. D. C 5. A 6. C 7. C 8. D 9. C. D. C. B. A. D 5. A 6. B 7. D 8. A 9. D. E. D. B. E. E 5. E.

### MATH 1242 FINAL EXAM Spring,

MATH 242 FINAL EXAM Spring, 200 Part I (MULTIPLE CHOICE, NO CALCULATORS).. Find 2 4x3 dx. (a) 28 (b) 5 (c) 0 (d) 36 (e) 7 2. Find 2 cos t dt. (a) 2 sin t + C (b) 2 sin t + C (c) 2 cos t + C (d) 2 cos t

### Separable First-Order Equations

4 Separable First-Order Equations As we will see below, the notion of a differential equation being separable is a natural generalization of the notion of a first-order differential equation being directly