Differential Equations not solvable

[bubblewrap]

In the first image it shows the $\alpha^2-w_0^2<0$ situation whereas in the second image the situation is when $\alpha^2-w_0^2=0$.The problem is the book says to use $T_0=2\pi/w_0$ to determine diabetes but you can't do that when $\alpha^2-w_0^2=0$ because it can't be put into a cosine function. What do I do in this situation?

 

[BvU]

I don't see the second image. Fig 2. has $\alpha^2-\omega_0^2>0$, not = 0. But the picture doesn't look credible at all. It looks more like underdamped to me.

Your book makes a mess of things with "three types, depending on $\alpha^2-\omega_0^2>0$, < 0 or zero. These three types correspond to overdamped, critically damped and underdamped cases".

Confusing, to put it mildly:

$\alpha^2-\omega_0^2>0 \qquad \Rightarrow$ overdamped
$\alpha^2-\omega_0^2=0 \qquad \Rightarrow$ critically damped
$\alpha^2-\omega_0^2<0 \qquad \Rightarrow$ underdamped​

Check out a better text, e.g. here

And yes, in the case of critical damping there are no oscillations.

By the way, I don't hear the book saying to use $T_0$. Did you quote correctly ?

 

[bubblewrap]

I meant Exercise 2 not the Figure 2, sorry for not pointing it out. And $T_0$ is on a different page, uploaded below.

 

[BvU]

It looks as if they want you to make quite some assumptions for this exercise: reliable measurements to begin with. G₀ = 70. And no additional glucose dosing in between, ...

If glucose concentration goes from 95 via 65 to 75 that means something significant for T₀ . Agree ?

( Could such measurements be consistent with critical damping ? )

 

[bubblewrap]

The measurements can be consistent if the function is of the form $(a-bt)e^{ct}$ where $a$,$b$ and $c$ are positive. I don't know the actual meaning of $T_0$ but from the hint I guess it has got something to do with the level going down to normal again.

 

[BvU]

The measurements can be consistent if the function is of the form $(a−bt)e^{ct}$ where a,b and c are positive

(I suppose you mean c is negative) Oh ? How do you come to that conclusion ?

From http://www.uprh.edu/rbaretti/Glucose1.htm