the management of a company has decided that the level of its investment should not exceed C dollars. Furthermore, managament has 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 inveated. Formulate but do not solve the problem in terms of a differential equation
To formulate the problem in terms of a differential equation, we need to define the variables and relationships involved.
Let:
- C represent the maximum investment level in dollars.
- T represent the total capital invested in dollars.
- I represent the rate of net investment, i.e., the rate of change of the total capital invested.
According to the problem, the rate of net investment is proportional to the difference between C and the total capital invested (C - T). We can express this relationship mathematically as:
I ∝ (C - T)
Since I is proportional to (C - T), we introduce a proportionality constant k to get:
I = k(C - T)
Now, to convert this relation into a differential equation, we need to express the rate of change of the total capital invested (dT/dt) in terms of I. We know that the rate of change of total capital invested is equal to the rate of net investment:
dT/dt = I
Substituting the expression for I, we obtain:
dT/dt = k(C - T)
This is the differential equation that represents the management's decision regarding investment level and the rate of net investment. Note that the specific solution for T(t) will depend on the initial conditions and the value of the proportionality constant k.