y + p(t)y = g(t) for each t I, and that also satisfies the initial condition y(t 0 ) = y 0 where y 0 is an arbitrary prescribed initial value.
|
|
- Malcolm Mills
- 5 years ago
- Views:
Transcription
1 p1 Differences Between Linear and Nonlinear Equation Theorem 1: If the function p and g are continuous on an open interval I : α < t < β containing the point t = t 0, then there exists a unique function y = φ(t) that satisfies the differential equation y + p(t)y = g(t) for each t I, and that also satisfies the initial condition y(t 0 ) = y 0 where y 0 is an arbitrary prescribed initial value.
2 p2 Differences Between Linear and Nonlinear Equation Theorem 2:(Containing nonlinear differential equation) Let the functions f and f y be continuous in some rectangle α < t < β, γ < y < δ containing the point (t 0, y 0 ). Then, in some interval t 0 h < t < t 0 + h contained in α < t < β, there is a unique solution y = φ(t) of the initial value problem y = f(t, y), y(t 0 ) = y 0
3 p3 Example. Use Theorem 1 to find an interval in which the initial value problem ty + 2y = 4t 2, y(1) = 2 has a unique solution. Sol. Rewriting ty + 2y = 4t 2 in the form y + 2 t y = 4t. Thus, p(t) = 2 t is continuous only for t < 0 or t > 0, and g(t) = 4t is continuous for all t. Since t 0 = 1 (0, ); consequently, the initial value problem ty + 2y = 4t 2, y(1) = 2
4 p4: continuation p3 has a unique solution on the interval 0 < t < since Quiz: ty + 2y = 4t 2 y + 2 y = 4t, for t > 0. t 1. Determine(without solving the problem) an interval in which the solution of the initial value problem is certain to exist. (4 t 2 )y + 2ty = 3t 2, y(1) = 3
5 p5 Example. Use Theorem 2 to find an interval in which the initial value problem dy dx = 3x2 + 4x + 2, y(0) = 1 2(y 1) has a unique solution. Sol. f(x, y) = 3x2 +4x+2 2(y 1) and f y (x, y) = 3x2 +4x+2. Thus each of 2(y 1) 2 f and f y functions are continuous everywhere except on the line y = 1.
6 p6: continuation p5 Consequently, the initial value problem dy dx = 3x2 + 4x + 2, y(0) = 1 2(y 1) has a unique solution in some interval about x = 0. Note: Though f and f y are continuous on (, ) (, 1) = {(x, y) < x <, < y < 1} containing a point (0, 1). This does not necessarily mean that the solution exists for all x. Exercise. Find the unique solution, and verify that the initial value problem has a unique solution only for x > 2.
7 p7 Example. Consider the initial value problem y = y 1 3, y(0) = 0 for t 0. Apply Theorem 2 to this initial value problem and then solve the problem. Sol. f(x, y) = y 1/3 and f y = 1 3 y 2/3 = 1. Thus f is 3y 2/3 continuous everywhere, f y does not exist when y = 0 and not continuous at y = 0. Hence there is not a rectangle R containing (0, 0) such that f and f y are continuous on R. So Theorem 2 does not apply to this problem.
8 p8: continuation p7 y = y 1/3 is rewritten as y 1/3 dy dt = 1 (separable) y 1/3 dy = 1 dt 3 2 y2/3 = t + C and y = [ ] 2 3/2 [ 2 (t + C) or y = 3 (t + C) 3 ] 3/2 t = 0 y = so y = φ 1 (t) = ( 2 3 t)3/2, the function y = φ 2 (t) = ( 2 3 t)3/2, initial value problem. [ ] 2 3/2 (0 + C) = 0 C = 0 3 t 0 is a solution. On the other hand, t 0 is also a solution of the
9 p9: continuation p8 Moreover, y = ψ(t) = 0, t 0 is yet another solution. For an arbitrary positive t 0, the function { 0, if 0 t < t0 y = X(t) = ± [ 2 3 (t t 0) ] 3/2, if t t0 are continuous, differentiable, and are solutions of the initial value problem y = y 1 3, y(0) = 0. Hence this problem has an infinite family of solutions.
10 p10: continuation p9 Remark. The continuity of f does assure the existence of solutions, but not their uniqueness.
11 p11 Quiz 2. Solve the initial value problem y = t 2 y(1 + t 3 ), y(0) = y 0 and determine how the interval in which the solution exists depends on the initial value y 0.
12 p12 Compare the first linear equation y + p(t)y = g(t) with nonlinear differential equation, such as y = y 2 : Interval of Definition Remark. The solution of a linear equation y + p(t)y = g(t) subject to the initial condition y(t 0 ) = y 0, exists throughout any interval about t = t 0 in which the function p and g are continuous. Thus, vertical asymptotes or other discontinuous in the solution can occur only at points of discontinuity of p or g.
13 p13 On the other hand, for a nonlinear initial value problem satisfying the hypotheses of Theorem 2, the interval in which a solution exists may be difficult to determine. Example. Solve the initial value problem y = y 2, y(0) = 1 and determine the interval in which the solution exists. Sol. f(t, y) = y 2 and f y (t, y) = 2y are continuous everywhere. So, by Theorem 2, there exists a h > 0 such that the initial value problem has a unique solution in interval for t in (0 h, 0 + h).
14 p14: continuation p13 and y = y 2 dy dt = y2 y 2 dy = dt y 2 dy = t = 0 y = 1 0+C (separable) dt y 1 = t + C y = 1 t + C = 1 C = 1, thus y = 1 1 t is the solution of the given initial value problem. Clearly, 1 lim t 1 1 t =, the solution exists only in the interval < t < 1. However, there is no indication from the differential equation itself.
15 p15 General Solution Example. y = 1 t+c is solutions of the nonlinear equation y = y 2 for arbitrary number C. But y(t) = 0 is also a solution, implies y = 1 t+c is not general solution since there is no a vlue of C such that y(t) = 1 t+c = 0 for all t. Implicit Solution The situation for nonlinear equations is much less satisfactory( 令人滿意的 ). Usually, the best that we can hope for is to find an equation F (t, y) = 0 involving t and y that is satisfied by the solution y = φ(t).
16 p16 Graphical or Numerical Construction of Integral Curves Because of the difficulty in obtaining exact analytical solutions of nonlinear differential equations, methods that yield approximate solutions or other qualitative information about solutions are of correspondingly greater importance. Summary. The linear equation y + p(t)y = g(t) has several nice properties: 1. Assuming that the coefficients are continuous, there is a general solution, containing an arbitrary constant, that includes all solutions of the differential equation. A particular solution that satisfies a given initial condition can be picked out by choosing the proper value for the arbitrary constant.
17 p17 u(t)g(t)dt u(t) 2. There is an expression for the solution, y = or [ ] y = 1 t u(t) t 0 u(s)g(s)ds + c where u(t) = e p(t)dt. Moreover, although it involves two integrations, the expression is an explicit one for the solution y = φ(t) rather than an equation that defines φ implicitly. 3. The possible points of discontinuity, or singularities, of the solution can be identified (without solving the problem) merely by finding the points of discontinuity of the coefficients. Thus, if the coefficients are continuous for all t, then the soloution also exists and is continuous for all t. None of these statements is true, in general, of nonlinear equations.
18 p18 Exercise. 1. It is possible to solve a nonlinear equation by making a change of the dependent variable that converts it into a linear equation. The most important such equations has the form y + p(t)y = q(t)y n and is called a Bernoulli equation after Jakob Bernoulli. (a) Solve Bernoulli s equation when n = 0, when n = 1 (b) Show that if n 0, 1, then the substitution v = y n 1 reduces Bernoulli s equation to a linear equation. (Note: This method of solution was found by Leibniz in 1696 )
19 p19: continuation p18 2. Solve the following equations by using the substitution (a) y = ry ky 2, r > 0 and k > 0 (This equation is important in population dynamics) Ans: r y = k+cre rt (b) y = ɛy σy 3, ɛ > 0 and σ > 0. (This equation occurs in the study of the stability of fluid flow) [ Ans: y = ± ] 1/2 ɛ σ+cɛe 2ɛt
First Order Differential Equations Lecture 3
First Order Differential Equations Lecture 3 Dibyajyoti Deb 3.1. Outline of Lecture Differences Between Linear and Nonlinear Equations Exact Equations and Integrating Factors 3.. Differences between Linear
More informationChapter 2: First Order DE 2.4 Linear vs. Nonlinear DEs
Chapter 2: First Order DE 2.4 Linear vs. Nonlinear DEs First Order DE 2.4 Linear vs. Nonlinear DE We recall the general form of the First Oreder DEs (FODE): dy = f(t, y) (1) dt where f(t, y) is a function
More informationFirst 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
More informationMA108 ODE: Picard s Theorem
MA18 ODE: Picard s Theorem Preeti Raman IIT Bombay MA18 Existence and Uniqueness The IVP s that we have considered usually have unique solutions. This need not always be the case. MA18 Example Example:
More informationLecture Notes for Math 251: ODE and PDE. Lecture 7: 2.4 Differences Between Linear and Nonlinear Equations
Lecture Notes for Math 51: ODE and PDE. Lecture 7:.4 Differences Between Linear and Nonlinear Equations Shawn D. Ryan Spring 01 1 Existence and Uniqueness Last Time: We developed 1st Order ODE models for
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 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 informationHomogeneous Equations with Constant Coefficients
Homogeneous Equations with Constant Coefficients MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Spring 2018 General Second Order ODE Second order ODEs have the form
More informationLinear Independence and the Wronskian
Linear Independence and the Wronskian MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Spring 2018 Operator Notation Let functions p(t) and q(t) be continuous functions
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 informationMath Homework 3 Solutions. (1 y sin x) dx + (cos x) dy = 0. = sin x =
2.6 #10: Determine if the equation is exact. If so, solve it. Math 315-01 Homework 3 Solutions (1 y sin x) dx + (cos x) dy = 0 Solution: Let P (x, y) = 1 y sin x and Q(x, y) = cos x. Note P = sin x = Q
More informationMATH 308 Differential Equations
MATH 308 Differential Equations Summer, 2014, SET 5 JoungDong Kim Set 5: Section 3.1, 3.2 Chapter 3. Second Order Linear Equations. Section 3.1 Homogeneous Equations with Constant Coefficients. In this
More informationSecond Order Linear Equations
October 13, 2016 1 Second And Higher Order Linear Equations In first part of this chapter, we consider second order linear ordinary linear equations, i.e., a differential equation of the form L[y] = d
More informationSign the pledge. On my honor, I have neither given nor received unauthorized aid on this Exam : 11. a b c d e. 1. a b c d e. 2.
Math 258 Name: Final Exam Instructor: May 7, 2 Section: Calculators are NOT allowed. Do not remove this answer page you will return the whole exam. You will be allowed 2 hours to do the test. You may leave
More informationLecture 9. Systems of Two First Order Linear ODEs
Math 245 - Mathematics of Physics and Engineering I Lecture 9. Systems of Two First Order Linear ODEs January 30, 2012 Konstantin Zuev (USC) Math 245, Lecture 9 January 30, 2012 1 / 15 Agenda General Form
More informationORDINARY 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
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 informationMath 333 Exam 1. Name: On my honor, I have neither given nor received any unauthorized aid on this examination. Signature: Math 333: Diff Eq 1 Exam 1
Math 333 Exam 1 You have approximately one week to work on this exam. The exam is due at 5:00 pm on Thursday, February 28. No late exams will be accepted. During the exam, you are permitted to use your
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 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 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 informationAgenda Sections 2.4, 2.5
Agenda Sections 2.4, 2.5 Reminders Read 3.1, 3.2 Do problems for 2.4, 2.5 Homework 1 due Friday Midterm Exam I on 1/23 Lab on Friday (Shapiro 2054) Office hours Tues, Thurs 3-4:30 pm (5852 East Hall) Theorem:
More informationChain Rule. MATH 311, Calculus III. J. Robert Buchanan. Spring Department of Mathematics
3.33pt Chain Rule MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Spring 2019 Single Variable Chain Rule Suppose y = g(x) and z = f (y) then dz dx = d (f (g(x))) dx = f (g(x))g (x)
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 informationFirst Order Differential Equations
First Order Differential Equations 1 Finding Solutions 1.1 Linear Equations + p(t)y = g(t), y() = y. First Step: Compute the Integrating Factor. We want the integrating factor to satisfy µ (t) = p(t)µ(t).
More informationORDINARY DIFFERENTIAL EQUATIONS
ORDINARY DIFFERENTIAL EQUATIONS GABRIEL NAGY Mathematics Department, Michigan State University, East Lansing, MI, 48824. JANUARY 3, 25 Summary. This is an introduction to ordinary differential equations.
More informationLecture 16. Theory of Second Order Linear Homogeneous ODEs
Math 245 - Mathematics of Physics and Engineering I Lecture 16. Theory of Second Order Linear Homogeneous ODEs February 17, 2012 Konstantin Zuev (USC) Math 245, Lecture 16 February 17, 2012 1 / 12 Agenda
More informationLecture 2. Classification of Differential Equations and Method of Integrating Factors
Math 245 - Mathematics of Physics and Engineering I Lecture 2. Classification of Differential Equations and Method of Integrating Factors January 11, 2012 Konstantin Zuev (USC) Math 245, Lecture 2 January
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 information2tdt 1 y = t2 + C y = which implies C = 1 and the solution is y = 1
Lectures - Week 11 General First Order ODEs & Numerical Methods for IVPs In general, nonlinear problems are much more difficult to solve than linear ones. Unfortunately many phenomena exhibit nonlinear
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 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 informationLecture 19: Solving linear ODEs + separable techniques for nonlinear ODE s
Lecture 19: Solving linear ODEs + separable techniques for nonlinear ODE s Geoffrey Cowles Department of Fisheries Oceanography School for Marine Science and Technology University of Massachusetts-Dartmouth
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 informationSection 2.1 (First Order) Linear DEs; Method of Integrating Factors. General first order linear DEs Standard Form; y'(t) + p(t) y = g(t)
Section 2.1 (First Order) Linear DEs; Method of Integrating Factors Key Terms/Ideas: General first order linear DEs Standard Form; y'(t) + p(t) y = g(t) Integrating factor; a function μ(t) that transforms
More informationFirst Order Differential Equations f ( x,
Chapter d dx First Order Differential Equations f ( x, ).1 Linear Equations; Method of Integrating Factors Usuall the general first order linear equations has the form p( t ) g ( t ) (1) where pt () and
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 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 informationChapter 4: Higher Order Linear Equations
Chapter 4: Higher Order Linear Equations MATH 351 California State University, Northridge April 7, 2014 MATH 351 (Differential Equations) Ch 4 April 7, 2014 1 / 11 Sec. 4.1: General Theory of nth Order
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 informationMIDTERM 1 PRACTICE PROBLEM SOLUTIONS
MIDTERM 1 PRACTICE PROBLEM SOLUTIONS Problem 1. Give an example of: (a) an ODE of the form y (t) = f(y) such that all solutions with y(0) > 0 satisfy y(t) = +. lim t + (b) an ODE of the form y (t) = f(y)
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 informationMATH 353 LECTURE NOTES: WEEK 1 FIRST ORDER ODES
MATH 353 LECTURE NOTES: WEEK 1 FIRST ORDER ODES J. WONG (FALL 2017) What did we cover this week? Basic definitions: DEs, linear operators, homogeneous (linear) ODEs. Solution techniques for some classes
More informationz 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
More information1+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
More informationLecture 13 - Wednesday April 29th
Lecture 13 - Wednesday April 29th jacques@ucsdedu Key words: Systems of equations, Implicit differentiation Know how to do implicit differentiation, how to use implicit and inverse function theorems 131
More information25. Chain Rule. Now, f is a function of t only. Expand by multiplication:
25. Chain Rule The Chain Rule is present in all differentiation. If z = f(x, y) represents a two-variable function, then it is plausible to consider the cases when x and y may be functions of other variable(s).
More informationProblem 1 In each of the following problems find the general solution of the given differential
VI Problem 1 dt + 2dy 3y = 0; dt 9dy + 9y = 0. Problem 2 dt + dy 2y = 0, y(0) = 1, y (0) = 1; dt 2 y = 0, y( 2) = 1, y ( 2) = Problem 3 Find the solution of the initial value problem 2 d2 y dt 2 3dy dt
More informationORDINARY DIFFERENTIAL EQUATION: Introduction and First-Order Equations. David Levermore Department of Mathematics University of Maryland
ORDINARY DIFFERENTIAL EQUATION: Introduction and First-Order Equations David Levermore Department of Mathematics University of Maryland 7 September 2009 Because the presentation of this material in class
More informationORDINARY DIFFERENTIAL EQUATIONS: Introduction and First-Order Equations. David Levermore Department of Mathematics University of Maryland
ORDINARY DIFFERENTIAL EQUATIONS: Introduction and First-Order Equations David Levermore Department of Mathematics University of Maryland 1 February 2011 Because the presentation of this material in class
More informationExistence Theory: Green s Functions
Chapter 5 Existence Theory: Green s Functions In this chapter we describe a method for constructing a Green s Function The method outlined is formal (not rigorous) When we find a solution to a PDE by constructing
More informationA Concise Introduction to Ordinary Differential Equations. David Protas
A Concise Introduction to Ordinary Differential Equations David Protas A Concise Introduction to Ordinary Differential Equations 1 David Protas California State University, Northridge Please send any
More informationMATH 251 Examination I February 23, 2017 FORM A. Name: Student Number: Section:
MATH 251 Examination I February 23, 2017 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
More informationLecture 5: Single Step Methods
Lecture 5: Single Step Methods J.K. Ryan@tudelft.nl WI3097TU Delft Institute of Applied Mathematics Delft University of Technology 1 October 2012 () Single Step Methods 1 October 2012 1 / 44 Outline 1
More informationApplied Differential Equation. November 30, 2012
Applied Differential Equation November 3, Contents 5 System of First Order Linear Equations 5 Introduction and Review of matrices 5 Systems of Linear Algebraic Equations, Linear Independence, Eigenvalues,
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 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 informationFirst-Order ODEs. Chapter Separable Equations. We consider in this chapter differential equations of the form dy (1.1)
Chapter 1 First-Order ODEs We consider in this chapter differential equations of the form dy (1.1) = F (t, y), where F (t, y) is a known smooth function. We wish to solve for y(t). Equation (1.1) is called
More informationPolytechnic 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
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 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 informationHW2 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,
More informationµ = e R p(t)dt where C is an arbitrary constant. In the presence of an initial value condition
MATH 3860 REVIEW FOR FINAL EXAM The final exam will be comprehensive. It will cover materials from the following sections: 1.1-1.3; 2.1-2.2;2.4-2.6;3.1-3.7; 4.1-4.3;6.1-6.6; 7.1; 7.4-7.6; 7.8. The following
More informationSection 4.7: Variable-Coefficient Equations
Cauchy-Euler Equations Section 4.7: Variable-Coefficient Equations Before concluding our study of second-order linear DE s, let us summarize what we ve done. In Sections 4.2 and 4.3 we showed how to find
More informationA BRIEF INTRODUCTION INTO SOLVING DIFFERENTIAL EQUATIONS
MATTHIAS GERDTS A BRIEF INTRODUCTION INTO SOLVING DIFFERENTIAL EQUATIONS Universität der Bundeswehr München Addresse des Autors: Matthias Gerdts Institut für Mathematik und Rechneranwendung Universität
More information2. First Order Linear Equations and Bernoulli s Differential Equation
August 19, 2013 2-1 2. First Order Linear Equations and Bernoulli s Differential Equation First Order Linear Equations A differential equation of the form y + p(t)y = g(t) (1) is called a first order scalar
More informationComputational Neuroscience. Session 1-2
Computational Neuroscience. Session 1-2 Dr. Marco A Roque Sol 05/29/2018 Definitions Differential Equations A differential equation is any equation which contains derivatives, either ordinary or partial
More information2.1 Differential Equations and Solutions. Blerina Xhabli
2.1 Math 3331 Differential Equations 2.1 Differential Equations and Solutions Blerina Xhabli Department of Mathematics, University of Houston blerina@math.uh.edu math.uh.edu/ blerina/teaching.html Blerina
More informationSeries Solutions Near an Ordinary Point
Series Solutions Near an Ordinary Point MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Spring 2018 Ordinary Points (1 of 2) Consider the second order linear homogeneous
More information(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
More informationMath 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
More informationFIRST-ORDER ORDINARY DIFFERENTIAL EQUATIONS I: Introduction and Analytic Methods David Levermore Department of Mathematics University of Maryland
FIRST-ORDER ORDINARY DIFFERENTIAL EQUATIONS I: Introduction and Analytic Methods David Levermore Department of Mathematics University of Maryland 4 September 2012 Because the presentation of this material
More informationFirst order differential equations
First order differential equations Samy Tindel Purdue University Differential equations and linear algebra - MA 262 Taken from Differential equations and linear algebra by Goode and Annin Samy T. First
More informationSolutions to Homework 1, Introduction to Differential Equations, 3450: , Dr. Montero, Spring y(x) = ce 2x + e x
Solutions to Homewor 1, Introduction to Differential Equations, 3450:335-003, Dr. Montero, Spring 2009 problem 2. The problem says that the function yx = ce 2x + e x solves the ODE y + 2y = e x, and ass
More informationPreliminary check: are the terms that we are adding up go to zero or not? If not, proceed! If the terms a n are going to zero, pick another test.
Throughout these templates, let series. be a series. We hope to determine the convergence of this Divergence Test: If lim is not zero or does not exist, then the series diverges. Preliminary check: are
More informationA brief introduction to ordinary differential equations
Chapter 1 A brief introduction to ordinary differential equations 1.1 Introduction An ordinary differential equation (ode) is an equation that relates a function of one variable, y(t), with its derivative(s)
More informationLinear Second Order ODEs
Chapter 3 Linear Second Order ODEs In this chapter we study ODEs of the form (3.1) y + p(t)y + q(t)y = f(t), where p, q, and f are given functions. Since there are two derivatives, we might expect that
More informationThe first order quasi-linear PDEs
Chapter 2 The first order quasi-linear PDEs The first order quasi-linear PDEs have the following general form: F (x, u, Du) = 0, (2.1) where x = (x 1, x 2,, x 3 ) R n, u = u(x), Du is the gradient of u.
More informationLecture 4: Numerical solution of ordinary differential equations
Lecture 4: Numerical solution of ordinary differential equations Department of Mathematics, ETH Zürich General explicit one-step method: Consistency; Stability; Convergence. High-order methods: Taylor
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 informationMath 312 Lecture 3 (revised) Solving First Order Differential Equations: Separable and Linear Equations
Math 312 Lecture 3 (revised) Solving First Order Differential Equations: Separable and Linear Equations Warren Weckesser Department of Mathematics Colgate University 24 January 25 This lecture describes
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 informationHomework 9 - Solutions. Math 2177, Lecturer: Alena Erchenko
Homework 9 - Solutions Math 2177, Lecturer: Alena Erchenko 1. Classify the following differential equations (order, determine if it is linear or nonlinear, if it is linear, then determine if it is homogeneous
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 informationLecture 10 - Second order linear differential equations
Lecture 10 - Second order linear differential equations In the first part of the course, we studied differential equations of the general form: = f(t, y) In other words, is equal to some expression involving
More informationExercises Chapter II.
Page 64 Exercises Chapter II. 5. Let A = (1, 2) and B = ( 2, 6). Sketch vectors of the form X = c 1 A + c 2 B for various values of c 1 and c 2. Which vectors in R 2 can be written in this manner? B y
More information2.2 Separable Equations
2.2 Separable Equations Definition A first-order differential equation that can be written in the form Is said to be separable. Note: the variables of a separable equation can be written as Examples Solve
More informationC H A P T E R. Introduction 1.1
C H A P T E R Introduction For y > 3/2, the slopes are negative, therefore the solutions are decreasing For y < 3/2, the slopes are positive, hence the solutions are increasing The equilibrium solution
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 informationDefinition 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
More informationFirst-Order Ordinary Differntial Equations II: Autonomous Case. David Levermore Department of Mathematics University of Maryland.
First-Order Ordinary Differntial Equations II: Autonomous Case David Levermore Department of Mathematics University of Maryland 25 February 2009 These notes cover some of the material that we covered in
More informationMATH 2413 TEST ON CHAPTER 4 ANSWER ALL QUESTIONS. TIME 1.5 HRS.
MATH 1 TEST ON CHAPTER ANSWER ALL QUESTIONS. TIME 1. HRS. M1c Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Use the summation formulas to rewrite the
More informationPartial proof: y = ϕ 1 (t) is a solution to y + p(t)y = 0 implies. Thus y = cϕ 1 (t) is a solution to y + p(t)y = 0 since
Existence and Uniqueness for LINEAR DEs. Homogeneous: y (n) + p 1 (t)y (n 1) +...p n 1 (t)y + p n (t)y = 0 Non-homogeneous: g(t) 0 y (n) + p 1 (t)y (n 1) +...p n 1 (t)y + p n (t)y = g(t) 1st order LINEAR
More informationDynamical Systems. August 13, 2013
Dynamical Systems Joshua Wilde, revised by Isabel Tecu, Takeshi Suzuki and María José Boccardi August 13, 2013 Dynamical Systems are systems, described by one or more equations, that evolve over time.
More informationFirst Order Linear Ordinary Differential Equations
First Order Linear Ordinary Differential Equations The most general first order linear ODE is an equation of the form p t dy dt q t y t f t. 1 Herepqarecalledcoefficients f is referred to as the forcing
More information(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
More information2.1 How Do We Measure Speed? Student Notes HH6ed. Time (sec) Position (m)
2.1 How Do We Measure Speed? Student Notes HH6ed Part I: Using a table of values for a position function The table below represents the position of an object as a function of time. Use the table to answer
More informationHomework Solutions: , plus Substitutions
Homework Solutions: 2.-2.2, plus Substitutions Section 2. I have not included any drawings/direction fields. We can see them using Maple or by hand, so we ll be focusing on getting the analytic solutions
More informationDRAFT - Math 101 Lecture Note - Dr. Said Algarni
2 Limits 2.1 The Tangent Problems The word tangent is derived from the Latin word tangens, which means touching. A tangent line to a curve is a line that touches the curve and a secant line is a line that
More information. For each initial condition y(0) = y 0, there exists a. unique solution. In fact, given any point (x, y), there is a unique curve through this point,
1.2. Direction Fields: Graphical Representation of the ODE and its Solution Section Objective(s): Constructing Direction Fields. Interpreting Direction Fields. Definition 1.2.1. A first order ODE of the
More information