D

Step 1: The motion described involves a particle with an initial velocity \( u \), an acceleration due to a force represented as \( a = kx \), where \( k \) is a constant.
Step 2: We can use the kinematic equation to relate velocity and displacement: \( v^2 = u^2 + 2ax \). Substituting the expression for acceleration gives us \( v^2 = u^2 + 2kx^2 \).
Step 3: The options represent various forms of this basic kinematic relationship. Only Option D correctly resolves to align with the derived equation.
Therefore, the correct option is D.