Question

For the differential equation s" + bs' +8s=0, find the values of b that make the general solution overdamped, underdamped, or critically damped.
I've gotten a few solutions, but the program is saying that my first interval for each is incorrect.
My solutions:
Overdamped: (-infinity,-4sqrt(2)),(4sqrt(2),infinity)
Underdamped: (-4sqrt(2),0),(0,4sqrt(2))
Critically Damped: [-4sqrt(2),-4sqrt(2)],[4sqrt(2),4sqrt(2)]

Answer 1

ok so...
\[a s'' +b s'+c s=0 \\ \text{ characteristic equation is } \\ ar^2+br+c=0 \\ \text{ so we have an overdamping if } b^2>4ac \\ \text{ we have an underdamping if } b^2<4ac \\ \text{ we have a critical damping if } b^2=4ac\]
is this what you used?

Answer 2

\[\text{ overdamping if } b^2>32 \\ \text{ underdamping if } b^2<32 \\ \text{ critical damping if } b^2=32\]

Answer 3

\[\text{ Let } k \text{ be a constant greater than } 0 \\
b^2>a \text{ means } b \in (-\infty,-\sqrt{a}) \cup (\sqrt{a},\infty) \\
b^2<a \text{ means } b \in (-\sqrt{a},\sqrt{a}) \\
b^2=a \text{ means } b \in \{ -\sqrt{a}, \sqrt{a} \}\]

Answer 4

I believe so yes, I may have messed up on the first intervals though

Answer 5

oops all of those a's were suppose to be k's

Answer 6

because I didn't want to use a for two different things :p

Answer 7

I got you haha, let me see if maybe it would only be on interval on the last two

Answer 8

One*

Answer 9

like the last one only has two solution so I didn't choose interval style to represent my solutions

Answer 10

This is what I'm encountering

Answer 11

The last two ended up being only one interval... maybe the first one is only one?

Answer 12

Ah okay, they were all only one interval for some strange reason!

Answer 13

Thanks for your help haha

Answer 14

one interval for each ?

Answer 15

the second one I thought should be one interval

Answer 16

(-4 sqrt(2),4 sqrt(2))

Answer 17

but the other two look fine to me given your instructions

Answer 18

Yes, so the first one was (4sqrt(2),infinity), second was (0,4sqrt(2)), and third was [4sqrt(2),4sqrt(2)]

Answer 19

For some reason, they just ended up being positive values.

Answer 20

yeah it looks like they ignored the negative values ...

Answer 21

Yep, well I guess it's easier... weird that they didn't specify that though.

Answer 22

I'm going to do some more research...but that thing about that I posted about the overdamping b^2>4ac and so on... thing came from an MIT notes site

Answer 23

Yeah, I remember seeing that in lecture and it definitely seems to work, it's just pulled from the quadratic equation haha... the part inside the square root!

Answer 24

right

Answer 25

Thanks man, I appreciate your help

Answer 26

https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0CB0QFjAAahUKEwiYzJb3x_DIAhVIbz4KHSfTATg&url=http%3A%2F%2Focw.mit.edu%2Fcourses%2Fmathematics%2F18-03sc-differential-equations-fall-2011%2Funit-ii-second-order-constant-coefficient-linear-equations%2Fdamped-harmonic-oscillators%2FMIT18_03SCF11_s13_2text.pdf&usg=AFQjCNHyxLH-FhgsnINAzZDzQu5kmZmUTA&sig2=Ofx1TiMXIykpxQVaDEPUlQ
no problem you are the one who figured out what they were looking for :p

Answer 27

haha nice