 #1
ForceBoy
 47
 6
 Homework Statement:

"A hospital patient is fed glucose intravenously (directly to the bloodstream) at a rate of r units per minute. The body removes glucose from the bloodstream at a rate proportional to the amount Q(t) present in the bloodstream at time t. " (Finney; Weir; Giordano, 452)
A) write the differential eq.
B) Solve the diff. eq ##Q(0) = Q_0##
C) find the limit as t goes to infinity
 Relevant Equations:
 The chapter this problem is found in is one on separable differential equations
The rate at which glucose enters the bloodstream is ##r## units per minute so:
## \frac{dI}{dt} = r ##
The rate at which it leaves the body is:
##\frac {dE}{dt} = k Q(t) ##
Then the rate at which the glucose in the body changes is:
A) ## Q'(t) = \frac{dI}{dt} + \frac {dE}{dt} = r  k Q(t) ##
I don't see how this is a separable differential equation. I still try to solve it.
##\frac{dQ}{dt} + k Q = r ##
## Q e^{kt} = \int r e^{kt} dt ##
## Q = \frac{r}{e^{kt}}\frac{e^{kx}}{k} ##
B) ## Q(t) = \frac{r}{k} ##
This tells me that the glucose in the bloodstream at any point in time will be a constant. I know this is wrong. It would be appreciated if someone could point me onto the right path to solve this diff. eq. Thank you.
## \frac{dI}{dt} = r ##
The rate at which it leaves the body is:
##\frac {dE}{dt} = k Q(t) ##
Then the rate at which the glucose in the body changes is:
A) ## Q'(t) = \frac{dI}{dt} + \frac {dE}{dt} = r  k Q(t) ##
I don't see how this is a separable differential equation. I still try to solve it.
##\frac{dQ}{dt} + k Q = r ##
## Q e^{kt} = \int r e^{kt} dt ##
## Q = \frac{r}{e^{kt}}\frac{e^{kx}}{k} ##
B) ## Q(t) = \frac{r}{k} ##
This tells me that the glucose in the bloodstream at any point in time will be a constant. I know this is wrong. It would be appreciated if someone could point me onto the right path to solve this diff. eq. Thank you.