Mathematical Physics Vol 1

5.2 Series Solutions of Differential Equations

227

5.2 Series Solutions of Differential Equations

It is known from mathematical analysis that a homogeneous linear differential equation with constant coefficients can be solved by algebraic methods, and that the solutions are elementary functions. For example, consider a homogeneous second-order linear differential equation with constant coefficients: ay ′′ + by ′ + cy = 0 , (5.16) where a , b , c are constants, and y = y ( x ) . It is assumed that its solution has the form where α and C are constants. The constant α is determined from the condition that the assumed solution satisfies the initial equation. This condition, by substituting (5.17) into (5.16), is reduced to an algebraic (quadratic) equation. a α 2 + b α + c = 0 , (5.18) which has two solutions ( α 1 , α 2 ). These solutions, in the general case, are complex numbers. The final solution of the initial equation (obtained by applying the superposition principle) is y = Ce α x , (5.17)

α 1 x

α 2 x

y = C 1 e

+ C 2 e

(5.19)

.

In practice, cases when the coefficients of equation (5.16) are not constant, but rather depend on x , are more frequent. In addition, equations can also be inhomogeneous, which makes their solution even more difficult. The solutions of these differential equations are often functions that are not elementary. This chapter will outline some of them (the most commonly used).

5.2.1 Solutions of Differential Equations using Power Series

Power series are most often used to solve differential equations when the solution cannot be obtained in closed form. This method is natural and relatively simple. Namely, all functions that appear in the observed differential equation are developed into a power series of x − x 0 (see (5.8)), or, in the special case, of x ( x 0 =0). Then a solution in the form of a power series is assumed y = ∞ ∑ m = 0 a m ( x − x 0 ) m , and the corresponding derivatives are determined and substituted in the initial equation. Finally, by equating the coefficients next to the same powers of x , we obtain the unknown coefficients a m , and consequently a solution (in the form of a series). We will demonstrate this technique on the examples of Legendre 1 andBessel 2 equations. However, let us first state a theorem of importance for solving these equations.

1 Adrien-Marie Legendre (1752-1833), French mathematician. He made substantial contributions in the field of special functions, elliptic integrals, number theory and calculus of variations. His book Éléments de géométrie (1794) was very well known and had 12 editions in a period of less than 30 years. 2 Friedrich Wilhelm Bessel (1784-1846), German astronomer and mathematician. His work on Bessel functions appeared in 1826.