# Power series are often used to represent the solutions to differential equations. One example is…

Influence train are frequently used to delineate the solutions to differential equations. One copy is Bessel's differential equation o d²y dy + x + (x2 – a?) y = 0 t20? a dx² + x x where a is a invariable. Bessel's differential equation arises when studying the sounds cer- tain harmonious instruments such as drums effect as courteous in divers other settings in physics and engineering. Solutions to the differential equation are unconcealed as Bessel dutys. In what follows, appropriate a is an integer. (a) Suppose a influence train Emo Anx" satisfies an-2+(n? - a?)an=0 for all nz 2. What is the radius of assembly of the influence train? Hint: touch the well-balanced and odd conditions partially. (b) Suppose that y(x) is absorbed by a influence train 2n=o anx" which satisfies (n? – a?)an=0 for n=0,1 and An-2+(n– a?)an=0 for all n > 2. Show that this duty solves Bessel's differential equation.