Sample Solutions of Assignment 3 for MAT3270B: 2.8,2.3,2.5,2.7
|
|
- Brent Alexander
- 5 years ago
- Views:
Transcription
1 Sample Solutions of Assignment 3 for MAT327B: 2.8,2.3,2.5, Transform the given initial problem into an equivalent problem with the initial point at the origin (a). dt = t2 + y 2, y(1) = 2, (b). dt = 1 y3, y( 1) = 3 Answer: (a)let t = s + 1, y = w + 2, then dw = 1 dt ds = 1 the original problem can be written as dw ds = (s + 1)2 + (w + 2), w() =. (b)let t = s 1, y = w + 3, then dw = 1 dt ds = 1 the original problem can be written as dw ds = 1 + (w + 3)2, w() =. 2. Use the method of successive approximations to solve the given initial value problem:(1) Determine φ n (t); (2) Find the limit of φ n 1
2 2 (a). y = 2(y + 1), y() = (b). y = y + 1 t, y() = Answer: (a) If y = φ(t), then the corresponding integral equation is φ(t) = 2(φ(s) + 1)ds If the initial approximation is φ (t) =, then and φ 2 (t) = φ n (t) = φ 1 (t) = 2ds = 2t 2(2t + 1)ds = 2t + 2t 2 2(φ n 1 (s) + 1)ds = n 1 2 k t k k! lim φ n(t) = e 2t 1 n (b) If y = φ(t), then the corresponding integral equation is φ(t) = (φ(s) + 1 s)ds If the initial approximation is φ (t) =, then and φ n (t) = φ 1 (t) = φ 2 (t) = (1 s)ds = t 1 2! t (1 1 2 s)ds = t 1 3! t3 (φ n 1 (s) + 1 s)ds = t tn+1 (n + 1)! lim n φ n(t) = t 3. A mass of.25 kg is dropped from rest in a medium offering a resistance of.2 v, where v is measured in m/sec.
3 (a)if the mass is dropped from a height of 3m, find its velocity when it hits the ground. (b)if the mass is to attain a velocity of no more than 1m/sec, find the maximum height from which it can dropped. (c)suppose that the resistance force is k v, where v is measured in m/sec and k is a constant. If the mass is dropped from a height of 3m and must hit the ground with velocity of no more than 1m/sec, determine the coefficient of the resistance k that is required. 3 Answer: Denote the mass m, the height of the mass x(t),the velocity v = x, the resistance force k v,, the initial height h, then, x () =, x() = h. By Newton s 2nd Law, we have mx = mg kx x () = x() = h Hence x + k m x = g e k m t (x + k m x ) = ge k m t (e k m t x ) = ge k m t e k m t x = m k g(e k m t 1) x = mg k (1 e k m t ) x = h mg k (t + m k (e k m t 1)) Here g = 9.8m/s 2. mg (a). From h = 3, i.e. (t + m k k (e k m t 1)) = 3, we have t = 3.635s.Then v = x = mg (1 k e k m t ) = m/s.
4 4 (b). x 1 mg k (1 e k m t ) 1 t s h m (c). According to the problem, x = h mg (t + m k k (e k m t 1)) v = x = mg (1 k e k m t ) x = h = 3 v = 1 which gives t = s and k = kg/s. Hence the coefficient of the resistance k must be kg/s. 4. Find the escape velocity for a bo projected upward with an initial velocity v from a point x = ξr above the surface of the earth, where R is the radius of the earth and ξ is a constant. Neglect the air resistance, find the initial altitude from which the bo must be launched in order to reduce the escape velocity to 85/1 of its value at the earth s surface. Answer: Denote the mass of the bo m, by Newton 2nd Law, we have mx = GMm x 2 x () = v x() = ξr
5 5 Here GM R 2 = g, i.e. GM = gr 2.Then mx = GMm x 2 x = GMx 2 2x x = 2GMx 2 x (x ) 2 v 2 = 2GM( 1 x 1 ξr ) Let x which means the bo can escape, we have (x ) 2 v 2 = 2GM ξr 2GM ξr v 2 = (x ) 2 + 2GM ξr Hence the escape velocity on the earth surface is 2GM. Assume the R escape velocity reduce 85% of its value on the earth surface, i.e. 2GM = ( 85 2GM )2 ξr 1 R ξ = R i.e. the initial altitude must be ( )R = R Suppose that a certain population has a growth rate that varies with time and that this population satisfies the differential equation dt = (.5 + sin (kt))y/n. If y() = y, find the time τ at which the population has doubled. Suppose y = 1, k = 2π, N = 5 estimate τ. Answer: The original equation can be written as then we get y = 1 (.5 + sin (kt))dt N y = ce 1 N (.5t 1 k cos (kt))
6 6 and c = y e 1 Nk by y() = y. Assuming y(τ) = 2y, and substituting y = 1, k = 2π, N = 5 to the above result, then Hence the τ 7.25 πτ cos 2πτ = 1π log (2e 1 1π ) 6. In the following problems, sketch the graph of f(y) versus y, determine the critical points and classify each one as asymptotically stable or unstable. (a). dt = y(y 1)(y 2), y (b). dt = (y 1)(ey 1), < y < + (c). dt = ay b y, a >, b >, y Answer: (To be continued)
7 t -1.5 Figure 1. for problem 6 f(y) = y(y 1)(y 2) b Figure 2. for problem 6 f(y) = (y 1)(e y 1); (a). From the figure of f(y) = y(y 1)(y 2), there are 3 critical points y =, y = 1, y = 2 and y = 1 is stable, y =, y = 2 are unstable. (b). From the figure of f(y) = (y 1)(e y 1), there are 2 critical points y =, y = 1 and y = is stable, y = 1 are unstable.
8 Figure 3. for problem 6 f(y) = ay b y. (c). From the figure of f(y) = ay b y, there are 2 critical points y =, y = b2 a 2 and y = is stable, y = b2 a 2 is unstable. 7. Solve the Gompertz equation dt = ry log (K y ), y() = y > Answer: The Original ODE can be rewritten as y y = r ln K r ln y
9 9 Let w = ln y, we have w + rw = r ln K (e rt w) = re rt ln K e rt w w = (e rt 1) ln K w = (1 e rt ) ln K + w e rt y = y e rt K 1 e rt 8. Consider the following Schaefer s model in population namics: dt = r(1 y )y Ey K Suppose E < r. Find the equilibrium points and state if they are stable or unstable. Answer: Let f(y) = r(1 y )y Ey, then K f (y) = r(1 y K ) r y K E Setting f(y) =, then y =, or y = k(1 E ) r At the point y =, f () = r E >, this is an unstable equilibrium point. At the point y = k(1 E ), f () = E r <, this is a stable equilibrium r point. 9. Consider the following bifurcation equation dt = ɛx x3
10 1 Show that for ɛ <, there exists only one critical point which is asymptotically stable; while for ɛ >, there are three critical points, of which one is unstable and the other two are stable. Answer: (a).if ɛ <, then ɛx x 3 = x(x 2 + ɛ ) = has only one root,so there exist only one critical point x =. Since x >, ɛx x 3 < and x <, ɛx x 3 >, the critical point x = is stable. (b). If ɛ >, then ɛx x 3 = x(x ɛ)(x ɛ) = has 3 root,so there exist 3 critical points ɛ,, ɛ. Since x < ɛ, ɛx x 3 > and ɛ < x <, ɛx x 3 <, the critical point x = ɛ is stable. Since ɛ < x <, ɛx x 3 < and < x < ɛ, ɛx x 3 >, the critical point x = is unstable. Since x > ɛ, ɛx x 3 < and < x < ɛ, ɛx x 3 >, the critical point x = ɛ is stable. i.e. the critical point x = is unstable and the critical points x = ± ɛ is stable. 1. Solve the following Chemical Reactions equations dx dt = α(p x)(q x), x() = x Answer: By separate variable method, Case 1: p = q dx (p x)(q x) = αdt dx (x p) 2 = αdt 1 x p 1 x p = αdt x = p + x p α(x p)t+1
11 11 Case 2: p q 1 x p 1 x q dx = αdt (p x)(q x) ln( x p x q x q x p = (p q)αdt ) = (p q)αt x p = x p x q x q e(p q)αt x = qαe(p q)αt (p x ) p(q x ) αe (p q)αt (p x ) (q x ) 11. Use Euler s method to find approximate values of the solution of the given initial value problem at t =.5, 1, 1.5, 2, 2.5, 3 with h =.1 (a). y = 5 3 y, y() = 2 (b). y = ty +.1y 3, y() = 1 Answer: (a) Setting f(t, y) = 5 3 y t =, y = 2, f = f(, 2) =.757 y 1 = y + f h = = Setting f n = f(t n, y n ), y n+1 = y n +.1f n. Hence, we get the following results: y(.5) = y 5 = 2.38 y(1) = y 1 = y(1.5) = y 15 = y(2) = y 2 = y(2.5) = y 25 = y(3) = y 3 =
12 12 (b) Setting f(t, y) = ty +.1y 3 t =, y = 1, f = f(, 2) =.1 y 1 = y + f h = = 1.1 Setting f n = f(t n, y n ), y n+1 = y n +.1f n. Hence, we get the following results: y(.5) = y 5 = y(1) = y 1 = y(1.5) = y 15 = y(2) = y 2 = y(2.5) = y 25 = y(3) = y 3 =
Math 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 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 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 informationSection , #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
More informationOld Math 330 Exams. David M. McClendon. Department of Mathematics Ferris State University
Old Math 330 Exams David M. McClendon Department of Mathematics Ferris State University Last updated to include exams from Fall 07 Contents Contents General information about these exams 3 Exams from Fall
More informationCalculus IV - HW 2 MA 214. Due 6/29
Calculus IV - HW 2 MA 214 Due 6/29 Section 2.5 1. (Problems 3 and 5 from B&D) The following problems involve differential equations of the form dy = f(y). For each, sketch the graph of f(y) versus y, determine
More informationEntrance Exam, Differential Equations April, (Solve exactly 6 out of the 8 problems) y + 2y + y cos(x 2 y) = 0, y(0) = 2, y (0) = 4.
Entrance Exam, Differential Equations April, 7 (Solve exactly 6 out of the 8 problems). Consider the following initial value problem: { y + y + y cos(x y) =, y() = y. Find all the values y such that the
More informationUse separation of variables to solve the following differential equations with given initial conditions. y 1 1 y ). y(y 1) = 1
Chapter 11 Differential Equations 11.1 Use separation of variables to solve the following differential equations with given initial conditions. (a) = 2ty, y(0) = 10 (b) = y(1 y), y(0) = 0.5, (Hint: 1 y(y
More informationMath Applied Differential Equations
Math 256 - Applied Differential Equations Notes Existence and Uniqueness The following theorem gives sufficient conditions for the existence and uniqueness of a solution to the IVP for first order nonlinear
More informationSolutions 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
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 informationMATH 307: Problem Set #3 Solutions
: Problem Set #3 Solutions Due on: May 3, 2015 Problem 1 Autonomous Equations Recall that an equilibrium solution of an autonomous equation is called stable if solutions lying on both sides of it tend
More informationHomework Solutions:
Homework Solutions: 1.1-1.3 Section 1.1: 1. Problems 1, 3, 5 In these problems, we want to compare and contrast the direction fields for the given (autonomous) differential equations of the form y = ay
More information88 Chapter 2. First Order Differential Equations. Problems 1 through 6 involve equations of the form dy/dt = f (y). In each problem sketch
88 Chapter 2. First Order Differential Equations place to permit successful breeding, and the population rapidl declined to extinction. The last survivor died in 1914. The precipitous decline in the passenger
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 informationMT410 EXAM 1 SAMPLE 1 İLKER S. YÜCE DECEMBER 13, 2010 QUESTION 1. SOLUTIONS OF SOME DIFFERENTIAL EQUATIONS. dy dt = 4y 5, y(0) = y 0 (1) dy 4y 5 =
MT EXAM SAMPLE İLKER S. YÜCE DECEMBER, SURNAME, NAME: QUESTION. SOLUTIONS OF SOME DIFFERENTIAL EQUATIONS where t. (A) Classify the given equation in (). = y, y() = y () (B) Solve the initial value problem.
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 informationHomework #4 Solutions
MAT 303 Spring 03 Problems Section.: 0,, Section.:, 6,, Section.3:,, 0,, 30 Homework # Solutions..0. Suppose that the fish population P(t) in a lake is attacked by a disease at time t = 0, with the result
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 informationAntiderivatives. Definition A function, F, is said to be an antiderivative of a function, f, on an interval, I, if. F x f x for all x I.
Antiderivatives Definition A function, F, is said to be an antiderivative of a function, f, on an interval, I, if F x f x for all x I. Theorem If F is an antiderivative of f on I, then every function of
More informationExam 2. May 21, 2008, 8:00am
PHYSICS 101: Fundamentals of Physics Exam 2 Exam 2 Name TA/ Section # May 21, 2008, 8:00am Recitation Time You have 1 hour to complete the exam. Please answer all questions clearly and completely, and
More informationBoyce/DiPrima/Meade 11 th ed, Ch 1.1: Basic Mathematical Models; Direction Fields
Boyce/DiPrima/Meade 11 th ed, Ch 1.1: Basic Mathematical Models; Direction Fields Elementary Differential Equations and Boundary Value Problems, 11 th edition, by William E. Boyce, Richard C. DiPrima,
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 informationReview of Lecture 5. F = GMm r 2. = m dv dt Expressed in terms of altitude x = r R, we have. mv dv dx = GMm. (R + x) 2. Max altitude. 2GM v 2 0 R.
Review of Lecture 5 Models could involve just one or two equations (e.g. orbit calculation), or hundreds of equations (as in climate modeling). To model a vertical cannon shot: F = GMm r 2 = m dv dt Expressed
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 informationMath Ordinary Differential Equations
Math 411 - Ordinary Differential Equations Review Notes - 1 1 - Basic Theory A first order ordinary differential equation has the form x = f(t, x) (11) Here x = dx/dt Given an initial data x(t 0 ) = x
More informationMath 210 Differential Equations Mock Final Dec *************************************************************** 1. Initial Value Problems
Math 210 Differential Equations Mock Final Dec. 2003 *************************************************************** 1. Initial Value Problems 1. Construct the explicit solution for the following initial
More information16.3 Conservative Vector Fields
16.3 Conservative Vector Fields Lukas Geyer Montana State University M273, Fall 2011 Lukas Geyer (MSU) 16.3 Conservative Vector Fields M273, Fall 2011 1 / 23 Fundamental Theorem for Conservative Vector
More informationSample Solutions of Assignment 9 for MAT3270B
Sample Solutions of Assignment 9 for MAT370B. For the following ODEs, find the eigenvalues and eigenvectors, and classify the critical point 0,0 type and determine whether it is stable, asymptotically
More informationSolutions for homework 1. 1 Introduction to Differential Equations
Solutions for homework 1 1 Introduction to Differential Equations 1.1 Differential Equation Models The phrase y is proportional to x implies that y is related to x via the equation y = kx, where k is a
More informationMath 307 E - Summer 2011 Pactice Mid-Term Exam June 18, Total 60
Math 307 E - Summer 011 Pactice Mid-Term Exam June 18, 011 Name: Student number: 1 10 10 3 10 4 10 5 10 6 10 Total 60 Complete all questions. You may use a scientific calculator during this examination.
More informationOrdinary Differential Equations: Worked Examples with Solutions. Edray Herber Goins Talitha Michal Washington
Ordinary Differential Equations: Worked Examples with Solutions Edray Herber Goins Talitha Michal Washington July 31, 2016 2 Contents I First Order Differential Equations 5 1 What is a Differential Equation?
More informationPOTENTIAL ENERGY AND ENERGY CONSERVATION
7 POTENTIAL ENERGY AND ENERGY CONSERVATION 7.. IDENTIFY: U grav = mgy so ΔU grav = mg( y y ) SET UP: + y is upward. EXECUTE: (a) ΔU = (75 kg)(9.8 m/s )(4 m 5 m) = +6.6 5 J (b) ΔU = (75 kg)(9.8 m/s )(35
More informationLecture Notes in Mathematics. Arkansas Tech University Department of Mathematics
Lecture Notes in Mathematics Arkansas Tech University Department of Mathematics Introductory Notes in Ordinary Differential Equations for Physical Sciences and Engineering Marcel B. Finan c All Rights
More informationLectures in Differential Equations
Lectures in Differential Equations David M. McClendon Department of Mathematics Ferris State University last revised December 2016 1 Contents Contents 2 1 First-order linear equations 4 1.1 What is a differential
More information= 2e t e 2t + ( e 2t )e 3t = 2e t e t = e t. Math 20D Final Review
Math D Final Review. Solve the differential equation in two ways, first using variation of parameters and then using undetermined coefficients: Corresponding homogenous equation: with characteristic equation
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 informationLecture 6, September 1, 2017
Engineering Mathematics Fall 07 Lecture 6, September, 07 Escape Velocity Suppose we have a planet (or any large near to spherical heavenly body) of radius R and acceleration of gravity at the surface of
More information9.3: Separable Equations
9.3: Separable Equations An equation is separable if one can move terms so that each side of the equation only contains 1 variable. Consider the 1st order equation = F (x, y). dx When F (x, y) = f (x)g(y),
More information(1 2t), y(1) = 2 y. dy dt = t. e t y, y(0) = 1. dr, r(1) = 2 (r = r(θ)) y = t(t2 + 1) 4y 3, y(0) = 1. 2t y + t 2 y, y(0) = 2. 2t 1 + 2y, y(2) = 0
MATH 307 Due: Problem 1 Text: 2.2.9-20 Solve the following initial value problems (this problem should mainly be a review of MATH 125). 1. y = (1 2t)y 2, y(0) = 1/6 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
More informationPHYSICS. Hence the velocity of the balloon as seen from the car is m/s towards NW.
PHYSICS. A balloon is moving horizontally in air with speed of 5 m/s towards north. A car is moving with 5 m/s towards east. If a person sitting inside the car sees the balloon, the velocity of the balloon
More informationMath 2214 Solution Test 1D Spring 2015
Math 2214 Solution Test 1D Spring 2015 Problem 1: A 600 gallon open top tank initially holds 300 gallons of fresh water. At t = 0, a brine solution containing 3 lbs of salt per gallon is poured into the
More informationFirst Order ODEs (cont). Modeling with First Order ODEs
First Order ODEs (cont). Modeling with First Order ODEs September 11 15, 2017 Bernoulli s ODEs Yuliya Gorb Definition A first order ODE is called a Bernoulli s equation iff it is written in the form y
More informationName Class. 5. Find the particular solution to given the general solution y C cos x and the. x 2 y
10 Differential Equations Test Form A 1. Find the general solution to the first order differential equation: y 1 yy 0. 1 (a) (b) ln y 1 y ln y 1 C y y C y 1 C y 1 y C. Find the general solution to the
More informationHomework #5 Solutions
Homework #5 Solutions Math 123: Mathematical Modeling, Spring 2019 Instructor: Dr. Doreen De Leon 1. Exercise 7.2.5. Stefan-Boltzmann s Law of Radiation states that the temperature change dt/ of a body
More informationChapter 9b: Numerical Methods for Calculus and Differential Equations. Initial-Value Problems Euler Method Time-Step Independence MATLAB ODE Solvers
Chapter 9b: Numerical Methods for Calculus and Differential Equations Initial-Value Problems Euler Method Time-Step Independence MATLAB ODE Solvers Acceleration Initial-Value Problems Consider a skydiver
More informationThe acceleration of gravity is constant (near the surface of the earth). So, for falling objects:
1. Become familiar with a definition of and terminology involved with differential equations Calculus - Santowski. Solve differential equations with and without initial conditions 3. Apply differential
More informationChapter 1: Introduction
Chapter 1: Introduction Definition: A differential equation is an equation involving the derivative of a function. If the function depends on a single variable, then only ordinary derivatives appear and
More informationMATH 4B Differential Equations, Fall 2016 Final Exam Study Guide
MATH 4B Differential Equations, Fall 2016 Final Exam Study Guide GENERAL INFORMATION AND FINAL EXAM RULES The exam will have a duration of 3 hours. No extra time will be given. Failing to submit your solutions
More informationFirst Order Differential Equations
C H A P T E R 2 First Order Differential Equations 2.1 5.(a) (b) If y() > 3, solutions eventually have positive slopes, and hence increase without bound. If y() 3, solutions have negative slopes and decrease
More informationPractice Exam 1 Solutions
Practice Exam 1 Solutions 1a. Let S be the region bounded by y = x 3, y = 1, and x. Find the area of S. What is the volume of the solid obtained by rotating S about the line y = 1? Area A = Volume 1 1
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 information20D - 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 =
More informationMATH 23 Exam 2 Review Solutions
MATH 23 Exam 2 Review Solutions Problem 1. Use the method of reduction of order to find a second solution of the given differential equation x 2 y (x 0.1875)y = 0, x > 0, y 1 (x) = x 1/4 e 2 x Solution
More informationOrdinary Differential Equation Theory
Part I Ordinary Differential Equation Theory 1 Introductory Theory An n th order ODE for y = y(t) has the form Usually it can be written F (t, y, y,.., y (n) ) = y (n) = f(t, y, y,.., y (n 1) ) (Implicit
More information34.3. Resisted Motion. Introduction. Prerequisites. Learning Outcomes
Resisted Motion 34.3 Introduction This Section returns to the simple models of projectiles considered in Section 34.1. It explores the magnitude of air resistance effects and the effects of including simple
More informationTutorial-1, MA 108 (Linear Algebra)
Tutorial-1, MA 108 (Linear Algebra) 1. Verify that the function is a solution of the differential equation on some interval, for any choice of the arbitrary constants appearing in the function. (a) y =
More informationODE Homework Solutions of Linear Homogeneous Equations; the Wronskian
ODE Homework 3 3.. Solutions of Linear Homogeneous Equations; the Wronskian 1. Verify that the functions y 1 (t = e t and y (t = te t are solutions of the differential equation y y + y = 0 Do they constitute
More informationMath 3301 Homework Set Points ( ) ( ) I ll leave it to you to verify that the eigenvalues and eigenvectors for this matrix are, ( ) ( ) ( ) ( )
#7. ( pts) I ll leave it to you to verify that the eigenvalues and eigenvectors for this matrix are, λ 5 λ 7 t t ce The general solution is then : 5 7 c c c x( 0) c c 9 9 c+ c c t 5t 7 e + e A sketch of
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 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 information144 Chapter 3. Second Order Linear Equations
144 Chapter 3. Second Order Linear Equations PROBLEMS In each of Problems 1 through 8 find the general solution of the given differential equation. 1. y + 2y 3y = 0 2. y + 3y + 2y = 0 3. 6y y y = 0 4.
More informationSection 2.1 Differential Equation and Solutions
Section 2.1 Differential Equation and Solutions Key Terms: Ordinary Differential Equation (ODE) Independent Variable Order of a DE Partial Differential Equation (PDE) Normal Form Solution General Solution
More information2nd-Order Linear Equations
4 2nd-Order Linear Equations 4.1 Linear Independence of Functions In linear algebra the notion of linear independence arises frequently in the context of vector spaces. If V is a vector space over the
More informationMATH165 Homework 7. Solutions
MATH165 Homework 7. Solutions March 23. 2010 Problem 1. 3.7.9 The stone is thrown with speed of 10m/s, and height is given by h = 10t 0.83t 2. (a) To find the velocity after 3 seconds, we first need to
More informationFinal Exam Review. Review of Systems of ODE. Differential Equations Lia Vas. 1. Find all the equilibrium points of the following systems.
Differential Equations Lia Vas Review of Systems of ODE Final Exam Review 1. Find all the equilibrium points of the following systems. (a) dx = x x xy (b) dx = x x xy = 0.5y y 0.5xy = 0.5y 0.5y 0.5xy.
More informationCalifornia State University Northridge MATH 280: Applied Differential Equations Midterm Exam 1
California State University Northridge MATH 280: Applied Differential Equations Midterm Exam 1 October 9, 2013. Duration: 75 Minutes. Instructor: Jing Li Student Name: Student number: Take your time to
More information14.9 Worked Examples. Example 14.2 Escape Velocity of Toro
14.9 Wored Examples Example 14. Escape Velocity of Toro The asteroid Toro, discovered in 1964, has a radius of about R = 5.0m and a mass of about m t =.0 10 15 g. Let s assume that Toro is a perfectly
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 informationAPPM 2360: Midterm exam 1 February 15, 2017
APPM 36: Midterm exam 1 February 15, 17 On the front of your Bluebook write: (1) your name, () your instructor s name, (3) your recitation section number and () a grading table. Text books, class notes,
More information6.5 Work and Fluid Forces
6.5 Work and Fluid Forces Work Work=Force Distance Work Work=Force Distance Units Force Distance Work Newton meter Joule (J) pound foot foot-pound (ft lb) Work Work=Force Distance Units Force Distance
More informationODE Math 3331 (Summer 2014) June 16, 2014
Page 1 of 12 Please go to the next page... Sample Midterm 1 ODE Math 3331 (Summer 2014) June 16, 2014 50 points 1. Find the solution of the following initial-value problem 1. Solution (S.O.V) dt = ty2,
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 informationExam 4 SCORE. MA 114 Exam 4 Spring Section and/or TA:
Exam 4 Name: Section and/or TA: Last Four Digits of Student ID: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may
More informationSMA 208: Ordinary differential equations I
SMA 208: Ordinary differential equations I First Order differential equations Lecturer: Dr. Philip Ngare (Contacts: pngare@uonbi.ac.ke, Tue 12-2 PM) School of Mathematics, University of Nairobi Feb 26,
More informationBessel s Equation. MATH 365 Ordinary Differential Equations. J. Robert Buchanan. Fall Department of Mathematics
Bessel s Equation MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Background Bessel s equation of order ν has the form where ν is a constant. x 2 y + xy
More informationSolution. It is evaluating the definite integral r 1 + r 4 dr. where you can replace r by any other variable.
Solutions of Sample Problems for First In-Class Exam Math 246, Fall 202, Professor David Levermore () (a) Give the integral being evaluated by the following MATLAB command. int( x/(+xˆ4), x,0,inf) Solution.
More informationMAT01B1: Separable Differential Equations
MAT01B1: Separable Differential Equations Dr Craig 3 October 2018 My details: acraig@uj.ac.za Consulting hours: Tomorrow 14h40 15h25 Friday 11h20 12h55 Office C-Ring 508 https://andrewcraigmaths.wordpress.com/
More informationLax Solution Part 4. October 27, 2016
Lax Solution Part 4 www.mathtuition88.com October 27, 2016 Textbook: Functional Analysis by Peter D. Lax Exercises: Ch 16: Q2 4. Ch 21: Q1, 2, 9, 10. Ch 28: 1, 5, 9, 10. 1 Chapter 16 Exercise 2 Let h =
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 informationThe First Derivative and Second Derivative Test
The First Derivative and Second Derivative Test James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University April 9, 2018 Outline 1 Extremal Values 2
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 informationModeling 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,
More information(1 y) 2 = u 2 = 1 u = 1
MATH 23 HOMEWORK #2 PART B SOLUTIONS Problem 2.5.7 (Semistable Equilibrium Solutions). Sometimes a constant equilibrium solution has the property that solutions lying on one side of the equilibrium solution
More informationMathematical Models. MATH 365 Ordinary Differential Equations. J. Robert Buchanan. Spring Department of Mathematics
Mathematical Models MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Spring 2018 Ordinary Differential Equations The topic of ordinary differential equations (ODEs)
More informationMathematical Models. MATH 365 Ordinary Differential Equations. J. Robert Buchanan. Fall Department of Mathematics
Mathematical Models MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Ordinary Differential Equations The topic of ordinary differential equations (ODEs) is
More informationThe First Derivative and Second Derivative Test
The First Derivative and Second Derivative Test James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University November 8, 2017 Outline Extremal Values The
More informationLaplace Transform Problems
AP Calculus BC Name: Laplace Transformation Day 3 2 January 206 Laplace Transform Problems Example problems using the Laplace Transform.. Solve the differential equation y! y = e t, with the initial value
More informationdy dt = 1 y t 1 +t 2 y dy = 1 +t 2 dt 1 2 y2 = 1 2 ln(1 +t2 ) +C, y = ln(1 +t 2 ) + 9.
Math 307A, Winter 2014 Midterm 1 Solutions Page 1 of 8 1. (10 points Solve the following initial value problem explicitly. Your answer should be a function in the form y = g(t, where there is no undetermined
More informationSolutions to Section 1.1
Solutions to Section True-False Review: FALSE A derivative must involve some derivative of the function y f(x), not necessarily the first derivative TRUE The initial conditions accompanying a differential
More informationENGI 2422 First Order ODEs - Separable Page 3-01
ENGI 4 First Order ODEs - Separable Page 3-0 3. Ordinary Differential Equations Equations involving only one independent variable and one or more dependent variables, together with their derivatives with
More informationHomework 3, due February 4, 2015 Math 307 G, J. Winter 2015
Homework 3, due February 4, 205 Math 307 G, J. Winter 205 In Problems -8, find the general solution of the equation. If initial conditions are given, solve also the initial value problem. is Problem. y
More informationModeling with First-Order Equations
Modeling with First-Order Equations MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Radioactive Decay Radioactive decay takes place continuously. The number
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 informationMath 216 First Midterm 18 October, 2018
Math 16 First Midterm 18 October, 018 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
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 informationExponential Growth (Doubling Time)
Exponential Growth (Doubling Time) 4 Exponential Growth (Doubling Time) Suppose we start with a single bacterium, which divides every hour. After one hour we have 2 bacteria, after two hours we have 2
More information1 y 2 dy = (2 + t)dt 1 y. = 2t + t2 2 + C 1 2t + t 2 /2 + C. 1 t 2 /2 + 2t 1. e y y = 2t 2. e y dy = 2t 2 dt. e y = 2 3 t3 + C. y = ln( 2 3 t3 + C).
Math 53 First Midterm Page. Solve each of the following initial value problems. (a) y = y + ty, y() = 3. The equation is separable : y = y ( + t). Thus y = y dy = ( + t)dt y = t + t + C t + t / + C. For
More informationMath 216 First Midterm 8 October, 2012
Math 216 First Midterm 8 October, 2012 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
More informationMATH 18.01, FALL PROBLEM SET #5 SOLUTIONS (PART II)
MATH 8, FALL 7 - PROBLEM SET #5 SOLUTIONS (PART II (Oct ; Antiderivatives; + + 3 7 points Recall that in pset 3A, you showed that (d/dx tanh x x Here, tanh (x denotes the inverse to the hyperbolic tangent
More information