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# Power Series Method

DOI link for Power Series Method

Power Series Method book

# Power Series Method

DOI link for Power Series Method

Power Series Method book

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## ABSTRACT

Solutions to differential equations that can be expressed exactly in terms of elementary functions like polynomials, rational functions, trigonometric functions, logarithmic functions, and so on are referred to as exact solutions. For example, the differential equation y y′ + =3 0 has y Ae x= −3 as its solution, and this is an exact solution. But, it may not always be possible to find such solutions to differential equations that arise in several applications. For the equation y xy′′ − = 0 (in literature, this equation is referred to as Airy’s differential equation), which is used to model the diffraction of light, it is not possible to find y x( ) that exactly satisfies the equation using the classical methods for solving ordinary differential equations (ODEs) (you may verify this). Hence, for such equations, we seek an approximate solution in terms of an infinite series that is arranged as powers of the independent variable. The series solution thus obtained is called a power series solution and is always a convergent series. The method of finding this form of solution is termed the power series method. This chapter describes two methods: 1. Algebraic or method of undetermined coefficients 2. Taylor series method for finding power series solutions to ODEs.