A charity show lasted for 4 hours and raised a total of $4000. It was assumed that the rate of raising the money was directly proportional as the time remaining and inversely proportional as the amount already raised.
(1) If x is the amount raised in time t, form a differential equation connecting x and t. /
(2) Solve this differential equation and show that the solution can be written in the form:

Question

A charity show lasted for 4 hours and raised a total of $4000. It was assumed that the rate of raising the money was directly proportional as the time remaining and inversely proportional as the amount already raised. 
(1) If x is the amount raised in time t, form a differential equation connecting x and t. /
(2) Solve this differential equation and show that the solution can be written in the form:

anonymous · Answer

$$\frac{ x^2 }{ (2000)^2 } + \frac{ (4-t)^2 }{ 16 } = 1$$

anonymous · Answer

I have done part (1). The equation I got is: $$\frac{ dx }{ dt } = \frac{ k(4-t) }{ x }$$
Please help me how do I find the value of k?