## Related questions with answers

Let f(t) denote the amount of capital invested by a certain business firm at time t. The rate of change of invested capital, $f^{\prime}(t)$, is sometimes called the rate of net investment. The management of the firm decides that the optimum level of investment should be C dollars and that, at any time, the rate of net investment should be proportional to the difference between C and the total capital invested. Construct a differential equation that describes this situation.

Solution

VerifiedIf $y=f(t)$ is the amount of capital invested by some business firm at time $t$, then the rate of net investment is $y'=f'(t)$. If the rate of net investment is supposed to be proportional to the difference between the optimum level of investment, $C$, and the total invested capital, then we can write the differential equation as

$\begin{aligned} \color{#4257b2}{y'=k(C-y)}, \end{aligned}$

where $k$ is initially a positive constant.

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