Question

Without actually solving the given differential equation, find the minimum radius of convergence R of power series solutions about the ordinary point x = 0. About the ordinary point x = 1. (x2 − 2x + 26)y'' + xy' − 4y = 0

Answer 1

Answer:

R1= 10.04

R2=10

Step-by-step explanation:

The differential equation is given by:

$(x^2 - 2x + 26)y'' + xy' - 4y = 0$

The minimum radius of convergence is given by the distance from the point x=a, to the nearest singularity points of the differential equation.

The singularity points are the roots of the polynomial that multiplies the second derivative of y. Thus, you have:

$x^2 - 2x + 26=0\\\\x_{1,2}=\frac{-(-2)\pm \sqrt{(-2)^2-4(1)(26)}}{2}=1\pm 10i$

the roots are in the complex plane. The radius of convergence is the distance between the points z1=0+0i, z2=1+0i (x=0 and x=1, respectively), in the complex plane, that is:

$R_1=|z_1-x_1|=\sqrt{1^2+10^2}=\sqrt{101}=10.04\\\\R_2=|z_2-x_1|=\sqrt{0^2+10^2}=\sqrt{100}=10$