Respuesta :
As per the given differential equation, the minimum radius of convergence R of power series solutions about the ordinary point x = 0 is 9 and that about the ordinary point x = 1 is 10 units.
Differential equation:
Differential equation refers the equation which contains one or more terms and the derivatives of one variable with respect to the other variable
And it can be written as,
=> dy/dx = f(x)
Given,
The given differential equation is (x² - 81)y"+ 7xy' + y = 0 R = 9 (x = 0) R = (x = 1).
Now we have to find the minimum radius of convergence R of power series solutions about the ordinary point x = 0 and x = 1.
Here we know that the minimum radius of convergence is defined as the distance between the ordinary point and the singularity of the differential equation.
And the Singularity point is the root of the polynomial attached with the second derivative. So, the singularity points will be calculated as,
=> x² - 81
=> (x + 9) (x - 9)
=> x = 9, - 9
So, the singularity points are 9, - 9.
So, the minimum radius of convergence can be calculated as,
Now, the ordinary points are 1 + 0i and 1 - 0i.
So,
=> r1 = | 1 + 0i - 9|
=> r1 = 9
=> r2 = | 1 + 0i + 9|
=> r2 = 10
Therefore, the minimum radius of convergence R of power series solutions about the ordinary point x = 0 is 9 and that about the ordinary point x = 1 is 10 units.
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