Introduction:
In this problem, we are given a body of mass 'm' that is subjected to a resistive force of magnitude 'b'. We are required to find the displacement 'T' as a function of time 't' in terms of the initial velocity 'vo' and the coefficient 'y = b/m'.

Explanation:
To solve this problem, we can start by applying Newton's second law of motion, which states that the force acting on an object is equal to the mass of the object multiplied by its acceleration.

Newton's Second Law:
F = ma

Since there is no restoring force in the medium, the only force acting on the body is the resistive force. Therefore, we can write the equation of motion as:

F = -bv

Where 'F' is the force, 'b' is the constant, and 'v' is the velocity of the body.

Equation of Motion:
-ma = -bv

Simplifying the equation, we get:

a = (b/m)v

Now, we can use the definition of acceleration to relate it to the derivative of velocity with respect to time:

a = dv/dt

Substituting this into the equation, we get:

dv/dt = (b/m)v

We can rearrange this differential equation by separating the variables:

dv/v = (b/m)dt

Integrating both sides of the equation, we get:

∫dv/v = (b/m)∫dt

ln|v| = (b/m)t + C

Where 'C' is the constant of integration.

Initial Condition:
At t = 0, the initial velocity is given by 'vo'. Therefore, we can substitute these values into the equation:

ln|vo| = (b/m)(0) + C

ln|vo| = C

Exponentiating both sides of the equation, we get:

|vo| = e^C

Since the magnitude of a velocity cannot be negative, we can drop the absolute value sign and write:

vo = e^C

Final Solution:
Substituting the value of 'C' into the equation, we get:

ln|v| = (b/m)t + ln|vo|

ln|v| = ln|vo| + (b/m)t

Taking the exponential of both sides of the equation, we get:

v = vo * e^(b/m)t

Finally, we can integrate the velocity with respect to time to find the displacement:

∫v dt = ∫vo * e^(b/m)t dt

T = (vo/b) * e^(b/m)t + D

Where 'T' represents the displacement as a function of time, 'vo' is the initial velocity, 'b' is the constant, 'm' is the mass, 'e' is the base of the natural logarithm, and 'D' is the constant of integration.