Using Power Series - Find The Solution of a Differential Equation

Question

xy''+2y'+xy=0

y(0)=1 and y'(0)=0

1) Find out where the first point on the positive-x-axis is located where the solution to the initial value problem is zero.

2) Find the closed form of the solution and find its interval of convergence.

Answer 1

The equations and formulas follow the LaTeX convention.

Let y=Σ_{n=0}a_nxⁿ. Then we have a₀=y(0)=1, a₁=y'(0)=0.

0=xy''+2y'+xy

=Σ_n=2n(n-1)a_nx^n-1+2Σ_n=1na_nx^n-1+Σ_n=0a_nxⁿ⁺¹

=Σn=1n(n+1)an+1xn+2Σn=0(n+1)an+1xn+Σn=1an-1xn

^=2a₁+(a₀+6a₂)x+Σ_n=2[(n+1)(n+2)a_n+1+a_n-1]xⁿ.

Therefore we must have the following recurrent formula:

a_n=a_n-2/n(n+1).

In particular, a_n=0 if n is odd, and

a_2k=(-1)^k/(2k+1)! for k=0,1,2,...

Thus y=sinx/x as x is nonzero and y(0)=1. The first point on the positive axis where y=0 is x=π.