ABSTRACT
The differential equations studied so far have all had closed form solutions, that is, their solutions could be expressed in terms of elementary functions, such as exponential, trigonometric, polynomial, and logarithmic functions, and most such elementary functions have expansions in terms of power series. However, there are a whole class of functions which are not elementary functions and which occur frequently in mathematical physics and engineering. The equations can sometimes be solved by discovering a power series that satisfies the differential equation, but the solution series may not be summable to an elementary function. Second-order ordinary differential equations that cannot be solved by analytical methods, that are those involving variable coefficients, can often be solved in the form of an infinite series of powers of the variable. It looks at some of the methods that make this possible — by the Leibniz— Maclaurin and Frobinius methods, involving Bessel's and Legendre's equations, Bessel and gamma functions and Legendre's polynomials.
