Solution:

Given differential equation: x^2(d^2y/dx^2) + 5x(dy/dx) + 4y = 0

To find the value of e^(2y(e)), we need to solve the given differential equation and find the expression for y(x).

Step 1: Convert the given differential equation into a standard form.
Let's divide the entire equation by x^2:
(d^2y/dx^2) + (5/x)(dy/dx) + (4/x^2)y = 0

Step 2: Identify the type of differential equation.
The given differential equation is a second-order linear homogeneous differential equation with variable coefficients.

Step 3: Solve the differential equation.
To solve the differential equation, we assume a solution of the form y(x) = x^r, where r is a constant.

Differentiating y(x) twice with respect to x, we get:
dy/dx = rx^(r-1)
d^2y/dx^2 = r(r-1)x^(r-2)

Substituting these derivatives into the differential equation, we get:
r(r-1)x^(r-2) + 5/x * rx^(r-1) + 4/x^2 * x^r = 0

Simplifying the equation, we get:
r(r-1)x^(r-2) + 5rx^(r-2) + 4x^(r-2) = 0

Factoring out x^(r-2), we get:
x^(r-2)(r(r-1) + 5r + 4) = 0

Since x > 0, the only way for the equation to hold true is when the expression inside the parentheses is equal to zero:
r(r-1) + 5r + 4 = 0

Simplifying the equation, we get:
r^2 + 4r + 4 = 0

Factorizing the equation, we get:
(r+2)^2 = 0

Taking the square root of both sides, we get:
r+2 = 0
r = -2

Step 4: Find the expression for y(x).
Since r = -2, the solution of the differential equation is y(x) = x^(-2).

Step 5: Use the initial conditions to find the value of the constant.
Given y(1) = 1, we substitute x = 1 and y = 1 into the expression for y(x):
1 = 1^(-2)
1 = 1

Given y'(1) = 0, we differentiate y(x) = x^(-2) with respect to x:
dy/dx = -2x^(-3)

Substituting x = 1 and y' = 0, we get:
0 = -2(1)^(-3)
0 = -2

Since the initial conditions lead to contradictory results, there is no solution that satisfies both conditions. Therefore, we cannot determine the value of e^(2y(e)).

In conclusion, the value of e^(2y(e)) cannot be determined as the given initial conditions lead to contradictory results.