Question

Determine the singular points of the differential equation

(x₁−11x+30)y′′−xy′+y=0

x₁= ; x₂= (Hint: Enter your answers such that x₁≤x₂)
Without actually solving the given DE, find a lower bound for the radius of convergence of power series solutions about the ordinary point x=4.
The lower bound for the radius of convergence of power series solutions is R= .

Answer 1

To determine the singular points of a differential equation, set the coefficients of y'' and y' equal to zero and solve for x₁ and x₂. The lower bound for the radius of convergence of power series solutions about an ordinary point can be found by determining the minimum distance between the ordinary point and the singular points.

Step-by-step explanation:

Differential equation:

The given differential equation is (x₁−11x+30)y′′−xy′+y=0. To determine the singular points, we need to find the values of x where the coefficients of y'' and y' become zero.

Steps:

1. Set the coefficients equal to zero and solve for x₁. 2. Repeat the same step for y' coefficient and solve for x₂. The singular points are x₁ and x₂.

Lower bound for radius of convergence:

To find the lower bound for the radius of convergence of power series solutions about the ordinary point x=4, we can use the formula |x-4| < R, where R is the radius of convergence. R should be greater than the distance between the ordinary point and the nearest singular point. So, the lower bound for R is the minimum distance between x=4 and the singular points x₁ and x₂.