On linear and non-linear equations.(sect. 2.4).
|
|
- Edward Newman
- 6 years ago
- Views:
Transcription
1 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 differential equations. Theorem Variable coefficients) Given continuous functions a, b : t, t 2 ) R, with t 2 > t, and given constants t 0 t, t 2 ), y 0 R, the IVP y = at) y + bt), yt 0 ) = y 0, has the unique solution y : t, t 2 ) R given by yt) = [ t ] y 0 + µs) bs) ds, ) µt) where the integrating factor function is given by t 0 µt) = e At), At) = t t 0 as) ds. Proof: Based on the integration factor method.
2 Review: Linear differential equations. Remarks: The Theorem above assumes that the coefficients a, b, are continuous in t, t 2 ) R. The Theorem above implies: a) There is an explicit expression for the solutions of a linear IVP, given in Eq. ). b) For every initial condition y 0 R there exists a unique solution to a linear IVP. c) For every initial condition y 0 R the corresponding solution yt) of a linear IVP is defined for all t t, t 2 ). None of these properties holds for solutions to non-linear differential equations. 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.
3 Non-linear differential equations. Definition An ordinary differential equation y t) = f t, yt)) is called non-linear iff the function t, u) f t, u) is non-linear in the second argument. a) The differential equation y t) = y 3 is non-linear, since the t) function f t, u) = t 2 /u 3 is non-linear in the second argument. b) The differential equation y t) = 2tyt) + ln yt) ) is non-linear, since the function f t, u) = 2tu + lnu) is non-linear in the second argument, due to the term lnu). c) The differential equation y t) yt) = 2t2 is linear, since the function f t, u) = 2t 2 u is linear in the second argument. t2 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.
4 Properties of solutions to non-linear ODE. Theorem Non-linear ODE) Fix a non-empty rectangle R = t, t 2 ) u, u 2 ) R 2 and fix a function f : R R denoted as t, u) f t, u). If the functions f and u f are continuous on R, and t 0, y 0 ) R, then there exists a smaller open rectangle ˆR R with t 0, y 0 ) ˆR such that the IVP y t) = f t, yt)), yt 0 ) = y 0 has a unique solution y on the set ˆR R 2. Remarks: i) There is no general explicit expression for the solution yt) to a non-linear ODE. ii) Non-uniqueness of solution to the IVP above may happen at points t, u) R 2 where u f is not continuous. iii) Changing the initial data y 0 may change the domain on the variable t where the solution yt) is defined. Properties of solutions to non-linear ODE. Given non-zero constants a, a 2, a 3, a 4, find every solution y of y = t 2 y 4 + a 4 y 3 + a 3 y 2 + a 2 y + a ). Solution: The ODE is separable. So first, rewrite the equation as y 4 + a 4 y 3 + a 3 y 2 + a 2 y + a ) y = t 2, then we integrate in t on both sides of the equation, y 4 + a 4 y 3 + a 3 y 2 + a 2 y + a ) y dt = t 2 dt + c. Introduce the substitution u = yt), so du = y t) dt, u 4 + a 4 u 3 + a 3 u 2 ) + a 2 u + a du = t 2 dt + c.
5 Properties of solutions to non-linear ODE. Given non-zero constants a, a 2, a 3, a 4, find every solution y of y = t 2 y 4 + a 4 y 3 + a 3 y 2 + a 2 y + a ). Solution: Recall: u 4 + a 4 u 3 + a 3 u 2 + a 2 u + a ) du = t 2 dt + c. Integrate, and in the result substitute back the function y: 5 y 5 t) + a 4 4 y 4 t) + a 3 3 y 3 t) + a 2 2 y 2 t) + a yt) = t3 3 + c. The solution is in implicit form. It is the root of a polynomial degree five. There is no formula for the roots of a general polynomial degree five or bigger. There is no explicit expression for solutions y of the ODE. Properties of solutions to non-linear ODE. Find every solution y of the initial value problem y t) = y /3 t), y0) = 0. Remark: The equation above is non-linear, separable, and the function f t, u) = u /3 has derivative u f = 3 so u f is not continuous at u = 0. u 2/3, The initial condition above is precisely where f is not continuous. Solution: There are two solutions to the IVP above: The first solution is y t) = 0.
6 Properties of solutions to non-linear ODE. Find every solution y of the initial value problem y t) = y /3 t), y0) = 0. Solution: The second solution is obtained as follows: [yt) ] /3 y t) dt = dt + c. Then, the substitution u = yt), with du = y t) dt, implies that u /3 du = dt + c 3 ] 2/3 yt) = t + c, 2[ yt) = [ 2 3 t + c) ] 3/2 0 = y0) = 2 3 c ) 3/2 c = 0. So, the second solution is: y 2 t) = 2 3 t ) 3/2. Recall y t) = 0. Properties of solutions to non-linear ODE. Find the solution y to the initial value problem y t) = y 2 t), y0) = y 0. Solution: This is a separable equation. So, y dt y 2 = dt + c y = t + c yt) = t + c. Using the initial condition in the expression above, y 0 = y0) = c c = y 0 yt) = y 0 t). This solution diverges at t = /y 0, so its domain is R {y 0 }. The solution domain depends on the values of the initial data y 0.
7 Properties of solutions to non-linear ODE. Summary: Linear ODE: a) There is an explicit expression for the solution of a linear IVP. b) For every initial condition y 0 R there exists a unique solution to a linear IVP. c) The domain of the solution of a linear IVP is defined for every initial condition y 0 R. Non-linear ODE: i) There is no general explicit expression for the solution yt) to a non-linear ODE. ii) Non-uniqueness of solution to a non-linear IVP may happen at points t, u) R 2 where u f is not continuous. iii) Changing the initial data y 0 may change the domain on the variable t where the solution yt) is defined. 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.
8 The Bernoulli equation. Remark: The Bernoulli equation is a non-linear differential equation that can be transformed into a linear differential equation. Definition Given functions p, q : R R and a real number n, the differential equation in the unknown function y : R R given by is called the Bernoulli equation. y + pt) y = qt) y n Theorem The function y : R R is a solution of the Bernoulli equation for y + pt) y = qt) y n, n, iff the function v = /y n ) is solution of the linear differential equation v n )pt) v = n )qt). The Bernoulli equation. Given arbitrary constants a 0 and b 0, find every solution of the differential equation y = a 0 y + b 0 y 3. Solution: This is a Bernoulli equation. Divide the equation by y 3, y y 3 = a 0 y 2 + b 0. Introduce the function v = /y 2, with derivative v = 2y /y 3 ), into the differential equation above, v 2 = a 0v+b 0 v = 2a 0 v 2b 0 v +2a 0 v = 2b 0.
9 The Bernoulli equation. Given arbitrary constants a 0 and b 0, find every solution of the differential equation y = a 0 y + b 0 y 3. Solution: Recall: v + 2a 0 v = 2b 0. The last equation is a linear differential equation for v. This equation can be solved using the integrating factor method. Multiply the equation by µt) = e 2a 0t, e 2a 0 t v ) = 2b0 e 2a 0t e 2a 0t v = b 0 a 0 e 2a 0t + c We obtain that v = c e 2a 0t b 0 a 0. Since v = /y 2, y 2 = c e 2a 0t b 0 y = ± a 0 c e 2a 0 t b ) 0 /2. a 0
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
More informationLinear Variable coefficient equations (Sect. 2.1) Review: Linear constant coefficient equations
Linear Variable coefficient equations (Sect. 2.1) Review: Linear constant coefficient equations. The Initial Value Problem. Linear variable coefficients equations. The Bernoulli equation: A nonlinear equation.
More informationLinear Variable coefficient equations (Sect. 1.2) Review: Linear constant coefficient equations
Linear Variable coefficient equations (Sect. 1.2) Review: Linear constant coefficient equations. The Initial Value Problem. Linear variable coefficients equations. The Bernoulli equation: A nonlinear equation.
More informationModeling with first order equations (Sect. 2.3). The mathematical modeling of natural processes.
Modeling with first order equations (Sect. 2.3). The mathematical modeling of natural processes. Main example: Salt in a water tank. The experimental device. The main equations. Analysis of the mathematical
More informationThe integrating factor method (Sect. 1.1)
The integrating factor method (Sect. 1.1) Overview of differential equations. Linear Ordinary Differential Equations. The integrating factor method. Constant coefficients. The Initial Value Problem. Overview
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 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 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 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 informationFirst 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 informationFind the Fourier series of the odd-periodic extension of the function f (x) = 1 for x ( 1, 0). Solution: The Fourier series is.
Review for Final Exam. Monday /09, :45-:45pm in CC-403. Exam is cumulative, -4 problems. 5 grading attempts per problem. Problems similar to homeworks. Integration and LT tables provided. No notes, no
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 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 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 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 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 informationAMATH 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
More informationOrdinary Differential Equation Introduction and Preliminaries
Ordinary Differential Equation Introduction and Preliminaries There are many branches of science and engineering where differential equations arise naturally. Now days, it finds applications in many areas
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 information1.2. Linear Variable Coefficient Equations. y + b "! = a y + b " Remark: The case b = 0 and a non-constant can be solved with the same idea as above.
1 12 Liner Vrible Coefficient Equtions Section Objective(s): Review: Constnt Coefficient Equtions Solving Vrible Coefficient Equtions The Integrting Fctor Method The Bernoulli Eqution 121 Review: Constnt
More informationThe Laplace Transform and the IVP (Sect. 6.2).
The Laplace Transform and the IVP (Sect..2). Solving differential equations using L ]. Homogeneous IVP. First, second, higher order equations. Non-homogeneous IVP. Recall: Partial fraction decompositions.
More informationFirst Order ODEs, Part II
Craig J. Sutton craig.j.sutton@dartmouth.edu Department of Mathematics Dartmouth College Math 23 Differential Equations Winter 2013 Outline Existence & Uniqueness Theorems 1 Existence & Uniqueness Theorems
More informationLecture Notes for Math 251: ODE and PDE. Lecture 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 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 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 informationSolutions to Math 53 Math 53 Practice Final
Solutions to Math 5 Math 5 Practice Final 20 points Consider the initial value problem y t 4yt = te t with y 0 = and y0 = 0 a 8 points Find the Laplace transform of the solution of this IVP b 8 points
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 informationLaplace Transforms Chapter 3
Laplace Transforms Important analytical method for solving linear ordinary differential equations. - Application to nonlinear ODEs? Must linearize first. Laplace transforms play a key role in important
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 informationNon-homogeneous equations (Sect. 3.6).
Non-homogeneous equations (Sect. 3.6). We study: y + p(t) y + q(t) y = f (t). Method of variation of parameters. Using the method in an example. The proof of the variation of parameter method. Using the
More informationSeries Solution of Linear Ordinary Differential Equations
Series Solution of Linear Ordinary Differential Equations Department of Mathematics IIT Guwahati Aim: To study methods for determining series expansions for solutions to linear ODE with variable coefficients.
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 informationOn separated solutions of logistic population equation with harvesting
On separated solutions of logistic population equation with harvesting Philip Korman Department of Mathematical Sciences University of Cincinnati Cincinnati Ohio 4522-25 Abstract We provide a surprising
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 informationSolving the Heat Equation (Sect. 10.5).
Solving the Heat Equation Sect. 1.5. Review: The Stationary Heat Equation. The Heat Equation. The Initial-Boundary Value Problem. The separation of variables method. An example of separation of variables.
More informationUnderstand the existence and uniqueness theorems and what they tell you about solutions to initial value problems.
Review Outline To review for the final, look over the following outline and look at problems from the book and on the old exam s and exam reviews to find problems about each of the following topics.. Basics
More informationSolutions Definition 2: a solution
Solutions As was stated before, one of the goals in this course is to solve, or find solutions of differential equations. In the next definition we consider the concept of a solution of an ordinary differential
More informationFirst 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
More informationmultistep methods Last modified: November 28, 2017 Recall that we are interested in the numerical solution of the initial value problem (IVP):
MATH 351 Fall 217 multistep methods http://www.phys.uconn.edu/ rozman/courses/m351_17f/ Last modified: November 28, 217 Recall that we are interested in the numerical solution of the initial value problem
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 informationS-Class. Differential Eqns MAP2302. Prof. JLF King Wedn. 30Jan2019
Differential Eqns MAP2302 S-Class Prof JLF King Wedn 30Jan209 Welcome Write DNE in a blank if the described object does not exist or if the indicated operation cannot be performed Write expressions unambiguously
More informationThe Laplace Transform (Sect. 4.1). The Laplace Transform (Sect. 4.1).
The Laplace Transform (Sect. 4.1). s of Laplace Transforms. The Laplace Transform (Sect. 4.1). s of Laplace Transforms. The definition of the Laplace Transform. Definition The function F : D F R is the
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 informationLecture 7 - Separable Equations
Lecture 7 - Separable Equations Separable equations is a very special type of differential equations where you can separate the terms involving only y on one side of the equation and terms involving only
More informationODEs Cathal Ormond 1
ODEs Cathal Ormond 2 1. Separable ODEs Contents 2. First Order ODEs 3. Linear ODEs 4. 5. 6. Chapter 1 Separable ODEs 1.1 Definition: An ODE An Ordinary Differential Equation (an ODE) is an equation whose
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 informationChapter1. Ordinary Differential Equations
Chapter1. Ordinary Differential Equations In the sciences and engineering, mathematical models are developed to aid in the understanding of physical phenomena. These models often yield an equation that
More informationSolutions to 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 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 informationFirst-Order Ordinary Differntial Equations: Classification and Linear Equations. David Levermore Department of Mathematics University of Maryland
First-Order Ordinary Differntial Equations: Classification and Linear Equations David Levermore Department of Mathematics University of Maryland 1 February 2009 These notes cover some of the material that
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 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 informationThe Method of Undetermined Coefficients.
The Method of Undetermined Coefficients. James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University May 24, 2017 Outline 1 Annihilators 2 Finding The
More informationThree major steps in modeling: Construction of the Model Analysis of the Model Comparison with Experiment or Observation
Section 2.3 Modeling : Key Terms: Three major steps in modeling: Construction of the Model Analysis of the Model Comparison with Experiment or Observation Mixing Problems Population Example Continuous
More informationDifferential Equations Class Notes
Differential Equations Class Notes Dan Wysocki Spring 213 Contents 1 Introduction 2 2 Classification of Differential Equations 6 2.1 Linear vs. Non-Linear.................................. 7 2.2 Seperable
More informationMath53: Ordinary Differential Equations Autumn 2004
Math53: Ordinary Differential Equations Autumn 2004 Unit 2 Summary Second- and Higher-Order Ordinary Differential Equations Extremely Important: Euler s formula Very Important: finding solutions to linear
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 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 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 informationy + 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.
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
More informationA population is modeled by the differential equation
Math 2, Winter 2016 Weekly Homework #8 Solutions 9.1.9. A population is modeled by the differential equation dt = 1.2 P 1 P ). 4200 a) For what values of P is the population increasing? P is increasing
More informationDiff. Eq. App.( ) Midterm 1 Solutions
Diff. Eq. App.(110.302) Midterm 1 Solutions Johns Hopkins University February 28, 2011 Problem 1.[3 15 = 45 points] Solve the following differential equations. (Hint: Identify the types of the equations
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 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 informationMATH 308 Differential Equations
MATH 308 Differential Equations Summer, 2014, SET 6 JoungDong Kim Set 6: Section 3.3, 3.4, 3.5, 3.6 Section 3.3 Complex Roots of the Characteristic Equation Recall that a second order ODE with constant
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 informationSeries 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
More informationMAT 285: Introduction to Differential Equations. James V. Lambers
MAT 285: Introduction to Differential Equations James V. Lambers April 2, 27 2 Contents Introduction 5. Some Basic Mathematical Models............................. 5.2 Solutions of Some Differential Equations.........................
More informationMATH 308 Differential Equations
MATH 308 Differential Equations Summer, 2014, SET 1 JoungDong Kim Set 1: Section 1.1, 1.2, 1.3, 2.1 Chapter 1. Introduction 1. Why do we study Differential Equation? Many of the principles, or laws, underlying
More informationMath 123, Week 9: Separable, First-Order Linear, and Substitution Methods. Section 1: Separable DEs
Math 123, Week 9: Separable, First-Order Linear, and Substitution Methods Section 1: Separable DEs We are finally to the point in the course where we can consider how to find solutions to differential
More informationM343 Homework 3 Enrique Areyan May 17, 2013
M343 Homework 3 Enrique Areyan May 17, 013 Section.6 3. Consider the equation: (3x xy + )dx + (6y x + 3)dy = 0. Let M(x, y) = 3x xy + and N(x, y) = 6y x + 3. Since: y = x = N We can conclude that this
More informationAnswers to Problem Set Number MIT (Fall 2005).
Answers to Problem Set Number 5. 18.305 MIT Fall 2005). D. Margetis and R. Rosales MIT, Math. Dept., Cambridge, MA 02139). November 23, 2005. Course TA: Nikos Savva, MIT, Dept. of Mathematics, Cambridge,
More informationPractice 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
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 informationLecture 2. Introduction to Differential Equations. Roman Kitsela. October 1, Roman Kitsela Lecture 2 October 1, / 25
Lecture 2 Introduction to Differential Equations Roman Kitsela October 1, 2018 Roman Kitsela Lecture 2 October 1, 2018 1 / 25 Quick announcements URL for the class website: http://www.math.ucsd.edu/~rkitsela/20d/
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 informationHigher Order Linear Equations
C H A P T E R 4 Higher Order Linear Equations 4.1 1. The differential equation is in standard form. Its coefficients, as well as the function g(t) = t, are continuous everywhere. Hence solutions are valid
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 informationSolutions of separable equations often are in an implicit form; that is, not solved for the dependent variable.
Section 2.2 Separable DEs Key Terms: Separable DEs; the DE can be rearranged so that all terms containing the independent variable are on one side of the equal sign and all terms containing the dependent
More informationOrdinary Differential Equations
Ordinary Differential Equations (MA102 Mathematics II) Shyamashree Upadhyay IIT Guwahati Shyamashree Upadhyay ( IIT Guwahati ) Ordinary Differential Equations 1 / 1 Books Shyamashree Upadhyay ( IIT Guwahati
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 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 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 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 informationOrdinary Differential Equations
Chapter 10 Ordinary Differential Equations 10.1 Introduction Relationship between rate of change of variables rather than variables themselves gives rise to differential equations. Mathematical formulation
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 informationFirst Order Linear DEs, Applications
Week #2 : First Order Linear DEs, Applications Goals: Classify first-order differential equations. Solve first-order linear differential equations. Use first-order linear DEs as models in application problems.
More informationPower series solutions for 2nd order linear ODE s (not necessarily with constant coefficients) a n z n. n=0
Lecture 22 Power series solutions for 2nd order linear ODE s (not necessarily with constant coefficients) Recall a few facts about power series: a n z n This series in z is centered at z 0. Here z can
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 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 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 informationكلية العلوم قسم الرياضيات المعادالت التفاضلية العادية
الجامعة اإلسالمية كلية العلوم غزة قسم الرياضيات المعادالت التفاضلية العادية Elementary differential equations and boundary value problems المحاضرون أ.د. رائد صالحة د. فاتن أبو شوقة 1 3 4 5 6 بسم هللا
More informationVANDERBILT UNIVERSITY. MATH 2610 ORDINARY DIFFERENTIAL EQUATIONS Practice for test 1 solutions
VANDERBILT UNIVERSITY MATH 2610 ORDINARY DIFFERENTIAL EQUATIONS Practice for test 1 solutions The first test will cover all material discussed up to (including) section 4.5. Important: The solutions below
More informationUC Berkeley Math 10B, Spring 2015: Midterm 2 Prof. Sturmfels, April 9, SOLUTIONS
UC Berkeley Math 10B, Spring 2015: Midterm 2 Prof. Sturmfels, April 9, SOLUTIONS 1. (5 points) You are a pollster for the 2016 presidential elections. You ask 0 random people whether they would vote for
More information1.5 First Order PDEs and Method of Characteristics
1.5. FIRST ORDER PDES AND METHOD OF CHARACTERISTICS 35 1.5 First Order PDEs and Method of Characteristics We finish this introductory chapter by discussing the solutions of some first order PDEs, more
More informationORDINARY DIFFERENTIAL EQUATIONS
ORDINARY DIFFERENTIAL EQUATIONS Basic concepts: Find y(x) where x is the independent and y the dependent varible, based on an equation involving x, y(x), y 0 (x),...e.g.: y 00 (x) = 1+y(x) y0 (x) 1+x or,
More informationKEY CONCEPTS. Factoring is the opposite of expanding.
KEY CONCEPTS Factoring is the opposite of expanding. To factor simple trinomials in the form x 2 + bx + c, find two numbers such that When you multiply them, their product (P) is equal to c When you add
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 information