D

Step 1: Given the initial velocity (u), acceleration (a), and the position (x) of the particle. The relationship connecting these variables is derived from the equations of motion.

Step 2: According to the problem, acceleration is given as a function of position, specifically a = kx.

Step 3: To determine the relationship between the final velocity (v) and position (x), we can use the kinematic equation:

$$ v^2 = u^2 + 2ax $$
Substituting for acceleration, we have:
$$ v^2 = u^2 + 2(kx)x $$

Step 4: This simplifies to the form, matching with our options:

Therefore, the correct relation is:

$$ v^2 = u^2 + 2kx^2 $$

Hence, the answer is option D.