Apply Theorem 11 to find a minimum value for the radius of c

Apply Theorem 1.1 to find a minimum value for the radius of convergence of a power series solution at x_0 of the following equations. (x + 1)y\" - 3xy + y = 0, x_0 = 1 xy\" + 2x/1 - x y\' - (sin^2 x) y = 0, x_0 = 0

Solution

Given

1. (x+1)y\'\'-3xy+y=0

In standard form this differential equation can be written as

Y\'\' - 3X/(X+1) Y\' + Y =0

The singular point for this equation is

x+1 = 0

x= -1

The distance from this singular point to the point x0=1 is x=-1

so the radius of convergence is greater than or equal to -1

r > = -1

2 . xy\'\' + 2x/(1-x) Y\' - (sin^2x)y = 0

Y\'\' + 2x/x(1-x)Y\' - sin^2x/x y = 0

the singular point for this equation is 1-x=0

x=1

The distance from this singular point from x0=0 is 1

so the radius of convergence is greater than or equal to 1

r>=1