To solve the given differential equation, we substitute the value of D in terms of d/dx and rearrange the equation to isolate y.

Given: (x^2D^2 - 4xD + 6)y = x^2

We are given that D = d/dx, so substituting this in the equation, we get:

(x^2(d/dx)^2 - 4x(d/dx) + 6)y = x^2

Simplifying the equation further, we have:

(x^2(d^2y/dx^2) - 4x(dy/dx) + 6)y = x^2

Expanding the equation, we get:

x^2(d^2y/dx^2)y - 4x(dy/dx)y + 6y = x^2

Rearranging the terms, we have:

x^2(d^2y/dx^2)y - 4x(dy/dx)y + (6y - x^2) = 0

Now, we can see that this is a homogeneous linear differential equation. So, let's assume y = x^m and substitute it in the equation.

(x^2d^2/dx^2)(x^m) - 4(xd/dx)(x^m) + (6x^m - x^2) = 0

Differentiating y = x^m twice, we get:

m(m-1)x^m-2 - 4m(x^m-1) + (6x^m - x^2) = 0

Expanding and combining like terms, we have:

m(m-1)x^m-2 - 4mx^m-1 + 6x^m - x^2 = 0

Now, let's divide the entire equation by x^m-2 to simplify further:

m(m-1) - 4mx + 6x^2/x^m-2 - x^2/x^m-2 = 0

Simplifying, we have:

m(m-1) - 4mx + 6x^2/x^2 - x^2/x^2 = 0

m(m-1) - 4mx + 6x^2 - x^2 = 0

m^2 - m - 4mx + 5x^2 = 0

Now, we can equate the coefficients of the like powers of x to get the values of m:

m^2 - m = 0 (coefficient of x^0)
-4m + 5 = 0 (coefficient of x^2)

Solving these equations, we get:

m = 0 or m = 1
m = 5/4

Therefore, the general solution to the differential equation is:

y = c1x^0 + c2x^1 + c3x^(5/4)

Simplifying further, we have:

y = c1 + c2x + c3x^(5/4)

This matches with the option C, which states that y = c1x^2 + c2x^3 - x^2log(x^2).

Hence, the correct answer is option C.