System with varying mass - String

[Microzero]

System with varying mass --- String

A string of mass M and length L is placed near a hole on the top of a horizontal and smooth table. A slight disturbance is given to one end of the string at time t = 0 so that it leaves the table through the hole. Assume the string on the table remains at rest while the remaining part is moving down. Denote the length of the string under the table as y and the speed of the string that is moving down by v.

Find:
1. v as a function of y
2. y as a function of t
3. the velocity of string when the whole string leaves the table
4. the time when the whole string leaves the table

[ Use the D.E. dy/dx + y P(x) = Q(x) to solve the problem. ]

I don't know how to do the first 2 questions. Please give me some ideas.
~ Thank you ~
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Here is my idea:
By considering the momentum
Fext= M dv/dt + (v-u) dM/dt -----★
(Resnick, Halliday--Physics 4th edt., Ch.9)
Fext= Mg , u=0 (remaining part rest at table)
dM/dt = ρdy/dt
∴ ★ becomes :
dv/dt + v(1/L)dy/dt = g

 

[Nate810]

integrating this equation, we get:v = gt + C where C is an integration constant. From the initial condition, we can obtain v = gt when y=0 Therefore, we get v = gt + C - (g/L)yt 1. v as a function of y : v = gt + C - (g/L)yt 2. y as a function of t : y = (1/L)(C/g - v/g)t 3. The velocity of string when the whole string leaves the table : v = gt + C 4. Time when the whole string leaves the table : t = (C/g)L

 

[ogb p]

(1/L)dy/dt = v-g
dy/dt = L(v-g)
y =∫L(v-g)dt
y = Lvt -1/2g t^2
v = dy/dt = L(dv/dt) - gt