SOLUTION: I have this question I am extremely confused, I know I need to use differential equations but do not know how to solve it, please if you could show me steps for solving as well as

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Question 1188315: I have this question I am extremely confused, I know I need to use differential equations but do not know how to solve it, please if you could show me steps for solving as well as the solution. Thanks in advanced here is the problem:
management of a company has decided that the level of its investment should not exceed C
dollars. Furthermore, management decided that the rate of net investment (the rate of change of the total capital invested) should be proportional to the difference between C
and the total capital invested. Formulate and solve the differential equation.
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
I don't think you are given enough information to solve it completely,
for you are not given any initial conditions.  You'd need more than you're
given for there is no way to know what the constant of proportionality k
would be.  I'll do what I can.  But I hit a dead end.

The differential equation comes from these words:

the rate of net investment (the rate of change of the total capital invested)
should be proportional to the difference between C and the total capital
invested

dy%2Fdt%22%22=%22%22k%28C-y%29

Multiply through by dt:

dy%22%22=%22%22k%28C-y%29dt

divide through by C-y

dy%2F%28C-y%29%22%22=%22%22kdt

Integrate both sides

int%28dy%2F%28C-y%29%29%22%22=%22%22k%2Aint%28dt%29

ln%28C-y%29%22%22=%22%22kt+%2B+c%5B1%5D

Change to exponential form:

C-y%22%22=%22%22e%5E%28kt+%2B+c%5B1%5D%29

C-y%22%22=%22%22e%5E%28kt%29%2Ae%5Ec%5B1%5D%29

e%5Ec%5B1%5D is just another constant c2

C-y%22%22=%22%22c%5B2%5De%5E%28kt%29%29

-y%22%22=%22%22-C%2Bc%5B2%5De%5E%28kt%29%29

y%22%22=%22%22C-c%5B2%5De%5E%28kt%29%29

Nothing else is given except

the level of its investment should not exceed C

y%22%22=%22%22C-c%5B2%5De%5E%28kt%29%29%22%22%3C=%22%22C

We have no more information to get rid of those constants.

Sorry, that's as far as I can go with it.

Edwin