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differential equations, jeffrey chasnov [pdf]

Introduction to Differential Equations by Jeffrey R. Chasnov

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Introduction to Differential Equations written by Jeffrey R. Chasnov cover the following topics of complex number.

  • 0. A short mathematical review
    The trigonometric functions, The exponential function and the natural logarithm, Definition of the derivative, Differentiating a combination of functions (The sum or difference rule, The product rule, The quotient rule, The chain rule, Differentiating elementary functions (The power rule, Trigonometric functions, Exponential and natural logarithm functions, Definition of the integral, The fundamental theorem of calculus, Definite and indefinite integrals, Indefinite integrals of elementary functions, Substitution, Integration by parts, Taylor series, Functions of several variables, Complex numbers
  • 1. Introduction to odes
    The simplest type of differential equation

  • 2. First-order odes
    The Euler method, Separable equations, Linear equations, Applications (Compound interest, Chemical reactions, Terminal velocity, Escape velocity, RC circuit, The logistic equation)

  • 3. Second-order odes, constant coefficients
    The Euler method, The principle of superposition, The Wronskian, Homogeneous odes, Real, distinct roots (vii

  • CONTENTS, Complex conjugate, distinct roots, Repeated roots), Inhomogeneous odes, First-order linear inhomogeneous odes revisited, Resonance, Damped resonance
  • 4. The Laplace transform
    Definition and properties, Solution of initial value problems, Heaviside and Dirac delta functions (Heaviside function, Dirac delta function, Discontinuous or impulsive terms)

  • 5. Series solutions
    Ordinary points, Regular singular points: Cauchy-Euler equations (Real, distinct roots, Complex conjugate roots, Repeated roots)

  • 6. Systems of equations
    Matrices, determinants and the eigenvalue problem, Coupled first-order equations (Two distinct real eigenvalues, Complex conjugate eigenvalues, Repeated eigenvalues with one eigenvector, Normal modes)

  • 7. Nonlinear differential equations
    Fixed points and stability (One dimension, Two dimensions), One-dimensional bifurcations (Saddle-node bifurcation, Transcritical bifurcation, Supercritical pitchfork bifurcation, Subcritical pitchfork bifurcation, Application: a mathematical model of a fishery), Two-dimensional bifurcations (Supercritical Hopf bifurcation, Subcritical Hopf bifurcation)

  • 8. Partial differential equations
    Derivation of the diffusion equation, Derivation of the wave equation, Fourier series, Fourier cosine and sine series, Solution of the diffusion equation (Homogeneous boundary conditions, Inhomogeneous boundary conditions, Pipe with closed ends), Solution of the wave equation (Plucked string, Hammered string, General initial conditions), The Laplace equation (Dirichlet problem for a rectangle, Dirichlet problem for a circle), The Schrödinger equation (Heuristic derivation of the Schrödinger equation, The time-independent Schrödinger equation, Particle in a one-dimensional box, The simple harmonic oscillator, Particle in a three-dimensional box, The hydrogen atom)

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