The problem you've provided involves solving the non-homogeneous second-order differential equation:

$4y'' - y = xe^{x/2},$

subject to the initial conditions:

$y(0) = 1, \quad y'(0) = 0.$

Additionally, the solution is expected to take the form:

$y = c_1 + c_2 \cos x + c_3 \sin x - \ln|\cos x| - \sin x \ln|\sec x + \tan x|.$

Step 1: Homogeneous Solution

First, solve the corresponding homogeneous equation:

$4y'' - y = 0.$

Rewriting this equation:

$y'' - \frac{1}{4}y = 0.$

The characteristic equation is:

$r^2 - \frac{1}{4} = 0,$

which gives:

$r = \pm \frac{1}{2}.$

Therefore, the general solution to the homogeneous equation is:

$y_h(x) = c_1 e^{x/2} + c_2 e^{-x/2}.$

Step 2: Particular Solution via Variation of Parameters

Now, we solve for a particular solution to the non-homogeneous equation using the method of variation of parameters. The right-hand side of the equation is $xe^{x/2}$, which suggests we will vary the constants in the homogeneous solution.

Let's assume the particular solution takes the form:

$y_p(x) = u_1(x) e^{x/2} + u_2(x) e^{-x/2},$

where $u_1(x)$ and $u_2(x)$ are functions to be determined. The