Can anyone help with Diff EQ?

"If in the exponential model for population growth, dy/dt=ry, the constant growth rate r is replaced by a growth rate r(1-y/k) that decreases linearly as the size of the population increases, we obtain the logistic model for population growth,

dy/dt = ry (1 - (y/k)),

in which K is referred to as the carrying capacity of the population. Sketch the graph of f, find the critical points, and determine whether each is asymptotically stable or unstable."


This should be really simple and straight forward (only looking for graphs of dy/dt vs t, phase line, and solution graph) but it is tripping me up a bit. I think I've found the critical points (y=0,k)? I'm also having an issue trying to graph this function. It's quadratic, is it not?

Damn, I was really hoping someone might understand differential equations.

Llayden said:

Damn, I was really hoping someone might understand differential equations.

Click to expand...

They're tought for anyone. I could have helped you if it was 20 years ago when I took calculus. I haven't used any if it since though, and just don't remember anymore.

Llayden said:

Damn, I was really hoping someone might understand differential equations.

Click to expand...

Yeah, it's been 25 years since I was in that class, and I actually failed it first time around. It is a tough class, much tougher than calculus.

It has been over 10 years since I've seen any Diff EQ. I'm VERY rusty. If I recall, that equation is not exactly solvable (in "closed form") precisely because it is non-linear, which is why you aren't being asked for a solution but only an analysis and graph.

I still have my textbooks but I'd have to seriously brush up to be able to tackle this. o.o Sorry to disappoint…

Well, thanks everyone. I got it figured out (you're right Sapphyre). Yeah, this class is tough. That's why I was trying to employ any and all resources. Hahaha! Thanks again, though. I'm just gonna keep plugging away.

I've been out of calc for a few years now and i'd have to do research on how to properly set it up. lol

It's a Bernoulli equation, you can linearize and solve it with the substitution u = y^(1-2) = 1/y
y' - ry = -r/k*y^2
turns into
u' + ru = r/k

Yeah, I know that now. Hahaha! I posted this towards the beginning of the class. Things are different now. I didn't even know about linearization then, kind of made it impossible without knowing that.

Heheh. One of my favorite memories of college relates to Diff Eq class: Our professor, who looked remarkably like Santa Claus, had just assigned some brutal homework and lectured us about cheating and/or working together on it. No sooner did we get out the door and into the hallway than I heard whispers of, "Let's meet in the <hall name> lobby tonight. I'll work on the odds, and you do the evens?" I turned around right away and offered, "How about we break it into threes?" And so we did, and we were still at it for almost two hours!

The herd mentality is strong in our class. I would wager that the entire class works in 4-5 different groups constantly. Ours is an early morning class, three days a week. After class at least half of us move up the hill to the one and only place that has the single Diff EQ tutor. It's funny seeing us all flood in there, it's the "math lab" in the math dept building. There are usually many other students scattered around looking for help with anything from adult basic maths to beginning calculus. The looks on their faces is priceless when they see us move in, slide tables together, and start filling a white board with ideas. Even the tutors shy away as most of them haven't gone that far.

I've enjoyed the coursework so far. It's been fairly straightforward. I posted this after our first assignment, we hadn't even learned what a differential equation was yet! Much less how to manipulate, translate, or solve one. It was a jump in the deep end. I think the prof did that as a quick weed out tool, the check and see if anyone had any deficiencies that would have a negative impact on them. Right now things are super simple, we've regressed into some linear algebra for the moment. My intuition tells me this is because we are probably going to be delving into systems of differential equations soon.