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MATH150-sug

# MATH150-sug - Introduction to ordinary dierential equations...

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Introduction to ordinary differential equations Lecture notes for MATH 150 J. R. Chasnov Department of Mathematics Hong Kong University of Science and Technology

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Contents 0 A short mathematical review 1 0.1 Trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . 1 0.2 Exponential function and natural logarithm . . . . . . . . . . . . 1 0.3 Definition of derivative . . . . . . . . . . . . . . . . . . . . . . . . 2 0.4 Differentiating a combination of functions . . . . . . . . . . . . . 2 0.4.1 The sum or difference rule . . . . . . . . . . . . . . . . . . 2 0.4.2 The product rule . . . . . . . . . . . . . . . . . . . . . . . 2 0.4.3 The quotient rule . . . . . . . . . . . . . . . . . . . . . . . 2 0.4.4 The chain rule . . . . . . . . . . . . . . . . . . . . . . . . 3 0.5 Differentiating elementary functions . . . . . . . . . . . . . . . . 3 0.5.1 Power rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 0.5.2 Trigonometric functions . . . . . . . . . . . . . . . . . . . 3 0.5.3 Exponential and natural logarithm functions . . . . . . . 3 0.6 Definition of integral . . . . . . . . . . . . . . . . . . . . . . . . . 3 0.7 Fundamental theorem of Calculus . . . . . . . . . . . . . . . . . . 4 0.8 Definite and indefinite integrals . . . . . . . . . . . . . . . . . . . 4 0.9 Indefinite integrals of elementary functions . . . . . . . . . . . . . 5 0.10 Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 0.11 Integration by parts . . . . . . . . . . . . . . . . . . . . . . . . . 6 0.12 Taylor series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 0.13 Complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1 Introduction to odes 11 1.1 Definition of an ode . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2 The simplest type of differential equation . . . . . . . . . . . . . 11 2 First-order differential equations 13 2.1 Separable equations . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Linear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3.1 Compound interest . . . . . . . . . . . . . . . . . . . . . . 18 2.3.2 Annual percentage yield (APY) . . . . . . . . . . . . . . . 19 2.3.3 Rule of 72 . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3.4 Compound interest with deposits or withdrawals . . . . . 20 2.3.5 Chemical reaction . . . . . . . . . . . . . . . . . . . . . . 21 2.3.6 Terminal velocity of falling mass . . . . . . . . . . . . . . 23 2.3.7 Escape velocity . . . . . . . . . . . . . . . . . . . . . . . . 24 2.3.8 Logistic equation . . . . . . . . . . . . . . . . . . . . . . . 25 iii

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3 Second-order linear differential equations with constant coeffi- cients 27 3.1 The principle of superposition . . . . . . . . . . . . . . . . . . . . 28 3.2 Second-order linear homogeneous ode with constant coefficients . 28 3.2.1 Real, distinct roots . . . . . . . . . . . . . . . . . . . . . . 29 3.2.2 Complex conjugate, distinct roots . . . . . . . . . . . . . 31 3.2.3 Repeated roots . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3 Second-order linear inhomogeneous ode . . . . . . . . . . . . . . 34 3.3.1 A digression back to first-order linear inhomogeneous ode’s 37 3.3.2 Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.3.3 Damped resonance . . . . . . . . . . . . . . . . . . . . . . 39 4 The Laplace transform 43 4.1 Definition and properties of the Laplace transform . . . . . . . . 43 4.2 Solution of initial value problems . . . . . . . . . . . . . . . . . . 44 4.3 Heaviside and Dirac delta functions . . . . . . . . . . . . . . . . . 48 4.3.1 Heaviside function . . . . . . . . . . . . . . . . . . . . . . 48 4.3.2 Dirac delta function . . . . . . . . . . . . . . . . . . . . . 49 4.4 Discontinuous and impulsive inhomogeneous terms . . . . . . . . 50 5 Series solutions of second-order linear homogeneous differential equations 53 5.1 Ordinary points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 5.2 Regular singular points: Euler equations . . . . . . . . . . . . . . 57 5.2.1 Real, distinct roots . . . . . . . . . . . . . . . . . . . . . . 59 5.2.2 Complex conjugate roots . . . . . . . . . . . . . . . . . . 59 5.2.3 Repeated roots . . . . . . . . . . . . . . . . . . . . . . . . 59 6 Systems of first-order linear equations 61 6.1 Determinants and the eigenvalue problem . . . . . . . . . . . . . 61 6.2 Two coupled first-order linear homogeneous differential equations 62 6.2.1 Real eigenvalues with two linearly independent eigenvectors 62 6.2.2 Complex conjugate eigenvalues . . . . . . . . . . . . . . . 66 6.2.3 Repeated eigenvalues with one eigenvector . . . . . . . . . 67 6.3 Normal modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 iv
Chapter 0 A short mathematical review 0.1 Trigonometric functions The Pythagorean trigonometric identity is given by sin 2 x + cos 2 x = 1 , and the addition theorems are sin( x + y ) = sin( x ) cos( y ) + cos( x ) sin( y ) , cos( x + y ) = cos( x ) cos( y ) - sin( x ) sin( y ) .

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