Solution:

Complementary Function:
The complementary function is the general solution of the homogeneous differential equation obtained by setting the RHS of the given differential equation to zero.

Homogeneous Differential Equation:
A differential equation is said to be homogeneous if all the terms in it contain either the dependent variable or its derivatives.

Given differential equation is x 2y’’-xy’ 2y= x logx
This is a homogeneous differential equation because all the terms in it contain either the dependent variable y or its derivatives.

To find the complementary function, we need to solve the homogeneous differential equation:
x 2y’’-xy’ 2y= 0

Let y = xk
Differentiating once, we get
y’ = kx^(k-1)
Differentiating twice, we get
y’’ = k(k-1)x^(k-2)

Substituting into the given equation, we get
x 2y’’-xy’ 2y= x^2k(k-1)-x^(k+1) k^2x^(2k-2)

Simplifying, we get
k(k-1)-k^2=0
k(k-2)=0

Therefore, the two solutions of the homogeneous differential equation are y1 = x^2 and y2 = 1.

Thus, the complementary function is the linear combination of these solutions:
y_c = c1 x^2 + c2

where c1 and c2 are constants.

Therefore, the complementary function of the given differential equation is y_c = c1 x^2 + c2.