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

Since the rate of change of 11ebbd4d_3eea_5cc0_8a0a_594b3d1dcbf1_TB8202_11 with respect to 11ebbd4d_3eea_5cc1_8a0a_03568d315047_TB8202_11 is proportional to 11ebbd4d_3eea_5cc2_8a0a_2b55cfd26303_TB8202_11, we can write:

d/dt (11ebbd4d_3eea_5cc0_8a0a_594b3d1dcbf1_TB8202_11) = k * 11ebbd4d_3eea_5cc2_8a0a_2b55cfd26303_TB8202_11

where k is the proportionality constant.

This is a first-order linear differential equation which can be solved using separation of variables.

Separating the variables and integrating, we get:

(1/11ebbd4d_3eea_5cc0_8a0a_594b3d1dcbf1_TB8202_11) d(11ebbd4d_3eea_5cc0_8a0a_594b3d1dcbf1_TB8202_11) = k dt * 11ebbd4d_3eea_5cc2_8a0a_2b55cfd26303_TB8202_11

ln|11ebbd4d_3eea_5cc0_8a0a_594b3d1dcbf1_TB8202_11| = k * t * 11ebbd4d_3eea_5cc2_8a0a_2b55cfd26303_TB8202_11 + C

where C is the constant of integration.

Solving for 11ebbd4d_3eea_5cc0_8a0a_594b3d1dcbf1_TB8202_11, we get:

11ebbd4d_3eea_5cc0_8a0a_594b3d1dcbf1_TB8202_11 = e^(k*t*11ebbd4d_3eea_5cc2_8a0a_2b55cfd26303_TB8202_11+C)

Since 11ebbd4d_3eea_5cc0_8a0a_594b3d1dcbf1_TB8202_11 is a positive quantity, we can write:

11ebbd4d_3eea_5cc0_8a0a_594b3d1dcbf1_TB8202_11 = A * e^(k*t*11ebbd4d_3eea_5cc2_8a0a_2b55cfd26303_TB8202_11)

where A is a positive constant (equal to e^C).

Thus, the solution to the given differential equation is:

11ebbd4d_3eea_5cc0_8a0a_594b3d1dcbf1_TB8202_11 = A * e^(k*t*11ebbd4d_3eea_5cc2_8a0a_2b55cfd26303_TB8202_11)

The best choice is C because it accounts for the proportionality constant and shows 11ebbd4d_3eea_5cc0_8a0a_594b3d1dcbf1_TB8202_11 is exponentially related to 11ebbd4d_3eea_5cc2_8a0a_2b55cfd26303_TB8202_11.

Write and Solve the Differential Equation That Models the Following