what do you know about separable differential equations?
Hey first time user and thread starter,
I have a calculus word problem that I need to do for assignment but I absolutely don't know where to start.
Problem:
A glucose solution is administered intravenously at a constant rate r. The glucose
is consumed in the bloodstrem at a rate proportional to its concentration at any time.
Then if G is the glucose concentration at any time,
dG/dt = r - kG
where k is the proportionality constant. Determine the glucose concentration as a
function of time if the initial concentration is G0. What is the expect concentration
after a long period of time?
I have read the conditions for posting and in no way want a complete solution (Against rules, and doesn't help me learn) I just want to know where to start or what to look at first, then I can hopefully figure it out myself.
As I already know how to do the majority of the calculus that is in my course, just not good with interpreting word problems.
Thanks, Any help be appreciated
Hovering 'above' that differential equation is an equation with no differentials in it, which is what you want.
You can think of the given equation as the result of doing 'implicit differentiation' on the 'higher' equation. (Higher and lower will make sense, at least, if like me you were taught to think of differentiation as 'down' and integration 'up'.)
And you can see that the differentiation was with respect to t.
As things stand, though, you can't integrate both sides with respect to t.
But dG/dt + kG = r has potential. You just need to multiply through by an 'integrating factor'.
Edit: D'oh! let's not do all that (now in spoiler).
Spoiler:
Let's 'separate' as Skeeter says. And here's a pic just in case it helps.
__________________________________________________ __________
Don't integrate - balloontegrate!
Balloon Calculus; standard integrals, derivatives and methods
Balloon Calculus Drawing with LaTeX and Asymptote!
Ok what it seems like from my conclusion is mostly to do with integration as you somewhat said which is kinda weird considering that we have not really covered it yet. Concerning the course I am doing I am not doing a full time maths degree or anything doing another course with one maths subject in it that seems to cover the majority of I guess the basics. So if I see somewhat confused by some of the stuff you explained we may not have covered or touched on it yet. So Skeeter to tell the truth I don't know much at all about "Separable Differential Equations", which I guess could make this harder to help me so.
Sorry in advance.
PS: That weird balloon diagram sorta confused me more
dG/dt = r - kG
=> dt/dG = 1/(r - kG)
=>
You are now expected to integrate, use the given initial condition, and solve for G. Then consider the limit of this solution as t --> +oo.
Note that the answer to "What is the expect concentration after a long period of time?" can be calculated by solving dG/dt = 0 ....