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4: Series Solutions

  • Page ID
    89123
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    "In most sciences one generation tears down what another has built and what one has established another undoes. In mathematics alone each generation adds a new story to the old structure." - Hermann Hankel (1839-1873)

    • 4.1: Introduction to Power Series
      As NOTED A FEW TIMES, not all differential equations have exact solutions. So, we need to resort to seeking approximate solutions, or solutions i the neighborhood of the initial value. Before describing these methods, we need to recall power series.
    • 4.2: Power Series Method
      IN THE LAST EXAMPLE WE WERE ABLE to use the initial condition to produce a series solution to the given differential equation. Even if we specified more general initial conditions, are there other ways to obtain series solutions? Can we find a general solution in the form of power series?
    • 4.3: Singular Points
      The power series method does not always give us the full general solution to a differential equation. Problems can arise when the differential equation has singular points. The simplest equations having singular points are Cauchy-Euler equations.
    • 4.4: The Frobenius Method
    • 4.5: Legendre Polynomials
      Legendre Polynomials are one of a set of classical orthogonal polynomials. These polynomials satisfy a second-order linear differential equation. This differential equation occurs naturally in the solution of initial boundary value problems in three dimensions which possess some spherical symmetry.
    • 4.6: Bessel Functions
      BESSEL FUNCTIONS ARISE IN MANY PROBLEMS in physics possessing cylindrical symmetry, such as the vibrations of circular drumheads and the radial modes in optical fibers. They also provide us with another orthogonal set of basis functions.
    • 4.7: Gamma Function
      A FUNCTION THAT OFTEN OCCURS IN THE STUDY OF SPECIAL FUNCTIONS is the Gamma function. We will need the Gamma function in the next section on Fourier-Bessel series.
    • 4.8: Hypergeometric Functions
      Hypergeometric functions are probably the most useful, but least understood, class of functions. They typically do not make it into the undergraduate curriculum and seldom in graduate curriculum. Most functions that you know can be expressed using hypergeometric functions.
    • 4.9: Problems


    This page titled 4: Series Solutions is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by Russell Herman via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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