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:

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

Answer 2

The minimum radius of convergence R of power series solutions about the ordinary point x = 0 is 10.0498, and that about the ordinary point x = 1 is 10 units.

Given information:

The given differential equation is $(x^2 - 2x + 26)y'' + xy' - 4y = 0$.

It is required to find the minimum radius of convergence R of power series solutions about the ordinary point x = 0 and x = 1.

Now, the minimum radius of convergence is defined as the distance between the ordinary point and the singularity of the differential equation.

Singularity point is the root of the polynomial attached with the second derivative. So, the singularity points will be calculated as,

$x^2 - 2x + 26=0\\x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\\x=\dfrac{2\pm\sqrt{(-2)^2-4\times 1\times26}}{2}\\x=1\pm\sqrt{-100}\\x=1\pm10i$

So, the singularity points are $x_1=1+10i$ and $x_2=1-10i$.

Now, the ordinary points are $z_1=0+0i$ and $z_2=1+0i$.

The minimum radius of convergence can be calculated as,

$r_1=|z_1-x_1|\\=|0+0i-1-10i|\\=\sqrt{101}\\=10.0498\\r_2=|z_2-x_1|\\=\sqrt{100}\\=10$

Therefore, the minimum radius of convergence R of power series solutions about the ordinary point x = 0 is 10.0498 and that about the ordinary point x = 1 is 10 units.

For more details, refer to the link:

https://brainly.com/question/18763238