Write and solve the differential equation that models the following verbal statement: The rate of change of with respect to is proportional to .

Question

Write and solve the differential equation that models the following verbal statement: The rate of change of  with respect to  is proportional to  .

Quizplus · Accepted Answer

Let y be the function of 11eaa595_652d_9e6e_a696_83ebb99603a9_TB4584_11_TB4584_11 with respect to 11eaa595_652d_c57f_a696_379f08952311_TB4584_11_TB4584_11. Then, we can write the differential equation as:
dy/dt = ky
where k is the proportionality constant. This is a first-order linear differential equation, and its solution is given by:
y = Ce^kt
where C is the constant of integration. Since the rate of change of y is proportional to y, we can expect to see exponential growth or decay depending on the sign of k. Therefore, we can eliminate choices A and B since they are not exponential functions.
Now, we need to determine the sign of k. If the proportional constant is positive, the function will grow exponentially, and if it is negative, the function will decay exponentially. However, we don't have enough information in the verbal statement to determine the sign of k.
Therefore, the best choice is C since it is an exponential function, and it does not make any assumptions about the sign of k.

Write and Solve the Differential Equation That Models the Following