Chapter 7.4, Problem 51E

To determine

To find multiple choice. And the models is dy/dt directly proportional to y.

Answer to Problem 51E

Only II and III are directly proportional. Option (D).

Explanation of Solution

Given information:

(I). $y=e^{kt}+C$

(II). $y=C{e}^{kt}$

(III). $y={28}^{kt}$

(A) I only

(B) II only

(C) I and II only

(D) II and III only

(E) I, II, and III.

Calculation: The required solution is obtained as,

Check each equation by finding the derivatives to see if they can be written in the form dy/dt = Ky.

I.

  $\begin{array}{l}y=e^{kt}+C\\ \frac{dy}{dt}=k{e}^{kt}\\ =k\left(y-C\right)\ne Ky\end{array}$

II.

  $\begin{array}{l}y=C{e}^{kt}\\ \frac{dy}{dt}=kC{e}^{kt}\\ =ky\end{array}$

III.

  $\begin{array}{l}y={28}^{kt}\\ \frac{dy}{dt}=k\cdot {28}^{kt}\ln 28\\ =k\ln 28y\end{array}$

Hence, only II and III are directly proportional. Option (D).