Problem 2-a

Problem 2 [30 pts] (Bessel's Equation and Bessel Functions) Bessel's equation is a second order differential equation that arises in a variety of applications including calculating heat conduction in a cylinder, establishing dynamics for floating bodies, solving the radial Schrodinger equation to determine the shape of electron orbitals, finding diffraction from helical objects (including DNA), and many more. The actual differential equation is 2 2 day + (22 - 03 )y = 0. It can be shown that there is no "nice" expression for solutions to this equation involving the common functions in Calculus (like polynomials exponentials, trigonometric functions, and logarithms). However, it is possible to use the differential equation to write down series representations for the solutions. In fact, when a = 1, it can be shown that the power series below is a solution to this equation. (-1) 4 JI (2) = 2 2k . ki! ( k + 1 )! 2 24+1 k=0 Since we do not have a "nice" formula for Ji(x), we cannot easily evaluate the function at a specific x-value, but we can try to approximate Ji(2) by using Taylor polynomials. A. [8 pts] Explain whether using higher order Taylor polynomials would in general provide better or worse approximations for J, (10) than lower order ones. Make sure to show any computations that you need to establish your conclusion