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

Power series are often used to represent the solutions to differential equations. One example is Bessel’s differential equation o d²y dy + x + (x2 – a?) y = 0 t20? a dx² + x x where a is a constant. Bessel’s differential equation arises when studying the sounds cer- tain musical instruments such as drums make as well in many other settings in physics and engineering. Solutions to the differential equation are known as Bessel functions. In what follows, assume a is an integer. (a) Suppose a power series Emo Anx” satisfies an-2+(n? – a?)an=0 for all nz 2. What is the radius of convergence of

View complete question »Power series are often used to represent the solutions to differential equations. One example is Bessel’s differential equation o d²y dy + x + (x2 – a?) y = 0 t20? a dx² + x x where a is a constant. Bessel’s differential equation arises when studying the sounds cer- tain musical instruments such as drums make as well in many other settings in physics and engineering. Solutions to the differential equation are known as Bessel functions. In what follows, assume a is an integer. (a) Suppose a power series Emo Anx” satisfies an-2+(n? – a?)an=0 for all nz 2. What is the radius of convergence of the power series? Hint: handle the even and odd terms separately. (b) Suppose that y(x) is given by a power series 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 function solves Bessel’s differential equation.

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