The rate of change of y with respect to t is 3 times the value of the quantity 2 less than y. Find an equation for y, given that y = 212 when t = 0.
You get:

A. y=212e^3t-2
B. y=214e^3t-2
C. y=210e^3t+2
D. y=212e^3t+2
E. y=210e^3t-2

Question

The rate of change of y with respect to t is 3 times the value of the quantity 2 less than  y. Find an equation for y, given that y = 212 when t = 0.
You get:

 		A.	y=212e^3t-2
 		B.	y=214e^3t-2
 		C.	y=210e^3t+2
 		D.	y=212e^3t+2
 		E.	y=210e^3t-2

anonymous · Answer

$$ \frac{dy}{dt} = 3 \times (y-2) $$$$\int \frac{dy}{y-2} = 3 \int dt$$$$\ln|y-2| = 3t + c$$$$y - 2 = e^{3t+c}$$$$y-2=c_1e^{3t}$$
Using the initial conditions we have:
$$212 - 2 = c_1e^{(0)}$$$$c_1 = 210$$
Substituting back in the equation, we have:
$$y = 210e^{3t} + 2$$
Option C