# 6: Power Series Solutions of Differential Equations

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##### Objectives
• Learn how to solve second order ODEs using series.
• Use the power series method to solve the Laguerre equation.

• 6.1: Introduction to Power Series Solutions of Differential Equations
Many important differential equations in physical chemistry are second order homogeneous linear differential equations, but do not have constant coefficients. The following examples are all important differential equations in the physical sciences: the Hermite equation, the Laguerre equation, and the Legendre equation.
• 6.2: The Power Series Method
The power series method is used to seek a power series solution to certain differential equations. In general, such a solution assumes a power series with unknown coefficients, then substitutes that solution into the differential equation to find a recurrence relation for the coefficients.
• 6.3: The Laguerre Equation
Some differential equations can only be solved with power series methods. One such example is the Laguerre equation. This differential equation is important in quantum mechanics because it is one of several equations that appear in the quantum mechanical description of the hydrogen atom. The solutions of the Laguerre equation are called the Laguerre polynomials, and together with the solutions of other differential equations, form the functions that describe the orbitals of the hydrogen atom.
• 6.4: Problems

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