Solution:

Given differential equation is (D^2-4D+4) y=e^2x sec^2X.

To solve this differential equation, we will use the method of auxiliary equation.

Finding Auxiliary Equation:

The auxiliary equation for the given differential equation is obtained by replacing D with r.

r^2-4r+4 = 0

(r-2)^2 = 0

r = 2 (repeated root)

General Solution of Homogeneous Equation:

The general solution of the homogeneous equation is given by:

y_h = c1 e^(2x) + c2 x e^(2x)

where c1 and c2 are arbitrary constants.

Particular Solution of Non-Homogeneous Equation:

To find the particular solution of the non-homogeneous equation, we will use the method of undetermined coefficients.

We assume that the particular solution has the form:

y_p = A e^(2x) sec^2(x)

where A is a constant to be determined.

Taking the first and second derivatives of y_p, we get:

y'_p = 2A e^(2x) sec^2(x) + 2A e^(2x) sec(x) tan(x)

y''_p = 4A e^(2x) sec^2(x) + 8A e^(2x) sec(x) tan(x) + 4A e^(2x) sec(x) tan^2(x)

Substituting these values in the differential equation, we get:

(4A e^(2x) sec^2(x) + 8A e^(2x) sec(x) tan(x) + 4A e^(2x) sec(x) tan^2(x)) - 4(2A e^(2x) sec^2(x) + 2A e^(2x) sec(x) tan(x)) + 4(A e^(2x) sec^2(x)) = e^(2x) sec^2(x)

Simplifying the above equation, we get:

4A e^(2x) tan^2(x) = e^(2x) sec^2(x)

A = 1/4

Therefore, the particular solution is:

y_p = 1/4 e^(2x) sec^2(x)

General Solution of Non-Homogeneous Equation:

The general solution of the non-homogeneous equation is given by:

y = y_h + y_p

y = c1 e^(2x) + c2 x e^(2x) + 1/4 e^(2x) sec^2(x)

where c1 and c2 are arbitrary constants.

Final Answer:

The general solution of the given differential equation is:

y = c1 e^(2x) + c2 x e^(2x) + 1/4 e^(2x) sec^2(x)