Explanation:
To solve the problem, we need to use the product rule and the quotient rule of differentiation.

Product Rule:
The product rule states that if we have two functions u(x) and v(x), the derivative of their product is given by:

(d/dx)(u(x)v(x)) = u(x)(dv/dx) + v(x)(du/dx)

Quotient Rule:
The quotient rule states that if we have two functions u(x) and v(x), the derivative of their quotient is given by:

(d/dx)(u(x)/v(x)) = [v(x)(du/dx) - u(x)(dv/dx)] / v(x)^2

Step 1:
We first need to find the first derivative of Y using the product rule and the quotient rule.

Y = 2x + 4/x

(d/dx)Y = (d/dx)(2x) + (d/dx)(4/x)

(d/dx)Y = 2 + (-4/x^2) [using quotient rule]

(d/dx)Y = 2 - 4/x^2

Step 2:
Next, we need to find the second derivative of Y using the product rule and the quotient rule.

(d²/dx²)Y = (d/dx)(2 - 4/x^2)

(d²/dx²)Y = 0 + (8/x^3) [using quotient rule]

(d²/dx²)Y = 8/x^3

Step 3:
We then need to find x(dy/dx) using the product rule.

x(dy/dx) = x(d/dx)Y

x(dy/dx) = x(2 - 4/x^2)

x(dy/dx) = 2x - 4/x

Step 4:
Lastly, we need to find Y.

Y = 2x + 4/x

Step 5:
Now that we have all the necessary derivatives, we can substitute them into the given equation:

x²(d²/dx²)Y + x(dy/dx) - Y = x²(8/x^3) + (2x - 4/x) - (2x + 4/x)

Simplifying the equation, we get:

Final Answer:
6/x - 8/x³