Bessel FunctionThe Bessel function of order 0 is J 0 ( x ) = ∑ k = 0 ∞ ( − 1 ) k x 2 k 2 2 k ( k ! ) 2 (a) Show that the series converges for all x . (b) Show that tire series is a solution of the differential equation x 2 J 0 n + x J 0 ' + x 2 J 0 = 0 . (c) Use a graphing utility to graph the polynomial composed of the first four terms of J 0 (d) Approximate ∫ 0 1 J 0 d x accurate to two decimal places.
Solution Summary: The author explains how to prove that the series is a solution of differential equation.
A point is an ordinary point of a non-homogeneous linear second order differential equation,
y"+p(x)y'+q(x)y=f(x)
is p, q, f are analytic at x0. Moreover, our theorem on the existence of power series solutions extends to such differential equations, ie, we can find power series solutions of non-homogeneous linear differential equations in the same manner as we have for homogeneous equations...almost. Find the series solution to the non-homogeneous initial value problem. Note: BEWARE THE THEOREM OF EQUIVALENT SERIES!!!
Find the first five nonzero terms of the series solution to:
y"+2y'+y=ex, y(0)=2, y'(0)=1
Find a power series representation for the function. (Center your power series representation at x = 0.)
f(x)=1/(7+x)
f(x) is a periodic function with period 2π where the value of f(x) in the interval <x < is : (in pict)
Use Dirichlet's theorem to find the value at which the Fourier series in (Expansion f(x) using Fourier series) converges when x = 0, x = ±π/2, x = ±π, x = ±2π
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