Q8E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

Find a minimum value for the radius of convergence of a power series solution about x₀.

(x²-5x+6) y"-3xy'-y=0;  x₀=0

Step-by-Step Solution

Answer

The minimum value for the radius of convergence is equal to the distance between 0 and 2. 

Mi [Radius of convergence] =2

Determine the formula for Differential equations:

A differential equation is an equation containing an unknown function and its derivatives.

Writing the formula for Differential equations

a₂(x)y"+a₁(x)y'+a₀(x)y=0

Writing the formula for Differential equations in the standard form,

y"+p(x)y'+q(x)y=0

Where, p(x)=a₁(x)/a₂(x) and q(x)=a₀(x)/a₂(x)

The minimum value for radius of convergence of the differential equation around x 0 :

Determine the minimum value for radius of convergence of the differential equation around x₀ as below.

(x²-5x+6) y"-3xy'-y=0

Writing the formula for Differential equations in the standard form,

y"-3x/(x²-5x+6) y' - 1/(x²-5x+6) y=0

By using the above equation,

p(x)= - 3x/(x²-5x+6)

q(x)= -1/(x²-5x+6) 

Determine the minimum value for radius:

The singular points of f(x) are the roots for equation (x²-5x+6).

x²-5x+6=(x-2) (x-3)

In the above equation, the singular points are 2 and 3,

Therefore, the point 0 is an ordinary point.

The minimum value for the radius of convergence is equal to the distance between 0 and 2. 

Mi [Radius of convergence] =2