The Computable Differential Equation Lecture ... - Bruce E. Shapiro

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194 Integrating Factors CHAPTER 10. APPENDIX ON ANALYTIC METHODS (MATH 280 IN A NUTSHELL) An integrating factor µ for the differential equation satisfies If M(t, y)dt + N(t, y)dy = 0 ∂(µ(t, y)M(t, y)) ∂y = P (t, y) = M y − N t N ∂(µ(t, y)N(t, y)) ∂t is only a function of t (and not of y) then µ(t) = e R P (t)dt is an integrating factor.If Q(t, y) = N t − M y M is only a function of y (and not of t) then µ(t) = e R Q(t)dt is an integrating factor. Homogeneous Equations An equation is homogeneous if has the form y ′ = f(y/t) To solve a homogeneous equation, make the substitution y = tz and rearrange the equation; the result is separable: Bernoulli Equations A Bernoulli equation has the form dz F (z) − z = dt t y ′ (t) + p(t)y = q(t)y n for some number n. To solve a Bernoulli equation, make the substitution u = y 1−n The resulting equation is linear and y(t) = [ ( ∫ 1 C + µ )] 1/(1−n) µ(t)(1 − n)q(t)dt where ( ∫ µ(t) = exp (1 − n) ) p(t)dt Math 582B, Spring 2007 California State University Northridge c○2007, B.E.Shapiro Last revised: May 23, 2007
CHAPTER 10. APPENDIX ON ANALYTIC METHODS (MATH 280 IN A NUTSHELL) 195 Second Order Homogeneous Linear Equation with Constant Coefficients To solve the differential equation ay ′′ + by ′ + cy = 0 find the roots of the characteristic equation ar 2 + br + c = 0 If the roots (real or complex) are distinct, then If the roots are repeated then y = Ae r 1t + Be r 2t y = (A + Bt)e rt Method of Undetermined Coefficients To solve the differential equation ay ′′ + by ′ + cy = f(t) where f(t) is a polynomial, exponential, or trigonometric function, or any product thereof, the solution is y = y H + y P where y H is the complete solution of the homogeneous equation ay ′′ + by ′ + cy = 0 To find y P make an educated guess based on the form form of f(t). The educated guess should be the product y P = P (t)S(t)E(t) where P (t) is a polynomial of the same order as in f(t). S(t) = r n (A sin rt+B cos rt) is present only if there are trig functions in rt in f(t), and n is the multiplicity of r as a root of the characteristic equation (n = 0 if r is not a root). E(t) = r n e rt is present only if there is an exponential in rt in f(t). If f(t) = f 1 (t) + f 2 (t) + · · · then solve each of the equations ay ′′ + by ′ + cy = f i (t) separately and add all of the particular solutions together to get the complete particular solution. c○2007, B.E.Shapiro Last revised: May 23, 2007 Math 582B, Spring 2007 California State University Northridge
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The Computable Differential Equatio
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Contents 1 Classifying The Problem
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CONTENTS v Timeline on Computable D
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Chapter 1 Classifying The Problem 1
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CHAPTER 1. CLASSIFYING THE PROBLEM
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Chapter 2 Successive Approximations
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CHAPTER 2. SUCCESSIVE APPROXIMATION
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Chapter 3 Approximate Solutions 3.1
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Chapter 5 Runge-Kutta Methods 5.1 T
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Chapter 6 Linear Multistep Methods
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CHAPTER 6. LINEAR MULTISTEP METHODS
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Chapter 7 Delay Differential Equati
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CHAPTER 7. DELAY DIFFERENTIAL EQUAT
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