Relationship between velocity and displacement
The given equation v = 2x + 1 represents the relationship between the velocity (v) of moving particles and their displacement (x). To find out how x varies with time (t), we need to understand the relationship between displacement and time.

Understanding the equation
The equation v = 2x + 1 suggests that the velocity of the particles is directly proportional to their displacement. The constant term 1 indicates an initial velocity of 1 unit when the displacement is zero.

Deriving the equation for displacement
To find how x varies with t, we can integrate the given equation with respect to displacement (x). Integrating both sides gives:

∫v dx = ∫(2x + 1) dx

Integrating the left side gives:

∫v dx = x^2 + C1

Integrating the right side gives:

∫(2x + 1) dx = x^2 + x + C2

Combining the results, we have:

x^2 + C1 = x^2 + x + C2

Simplifying the equation, we find:

C1 = x + C2

Adding initial conditions
Assuming the initial position (x = 0) corresponds to t = 0, we can substitute these values into the equation to find the value of the constant C1:

C1 = 0 + C2

Therefore, C1 = C2.

Final equation for displacement
With the constant C1 = C2, the equation for displacement (x) as a function of time (t) can be expressed as:

x = t + C

Here, C represents the constant of integration, which can be determined by providing the initial conditions.

Conclusion
The displacement (x) of moving particles varies with time (t) according to the equation x = t + C, where C is a constant determined by the initial conditions. The given relationship v = 2x + 1 between velocity and displacement helps establish this equation through integration. By understanding this relationship, we can analyze and predict the behavior of the particles in terms of their displacement over time.