Chapter 7.3, Problem 37E

To determine

To find : the expression for change at time t.

Answer to Problem 37E

the expression is $Q\left(t\right)=3-3e^{kt}$ and the limit value is $3$ .

Explanation of Solution

Given information :

The differential equation is $\frac{dQ}{dt}=k\left(3-3Q\right)$ ,

Calculation: to find the expression first write the equation in standard form,

  $\begin{array}{l}\frac{dQ}{dt}=k\left(3-Q\right)\\ \frac{dQ}{\left(3-Q\right)}=kdt\text{ [separate variables]}\\ {\displaystyle \int \frac{dQ}{\left(3-Q\right)}={\displaystyle \int kdt\text{ [integrate both side]}}}\\ \ln \left|\left(3-Q\right)\right|=kt+c\text{ [simplified integrate]}\\ 3-Q=e^{kt+c}\text{ [remove ln]}\\ 3-Q=3e^{kt}\text{ [}{e}^c=3]\\ Q\left(t\right)=3-3e^{kt}\text{ [simplify]}\end{array}$

So, the expression is $Q\left(t\right)=3-3e^{kt}$ .

From the expression it can be observed that The limiting value of the expression is $3$ .