anonymous
  • anonymous
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)]
Differential Equations
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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freckles
  • freckles
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?
freckles
  • freckles
\[\text{ overdamping if } b^2>32 \\ \text{ underdamping if } b^2<32 \\ \text{ critical damping if } b^2=32\]
freckles
  • freckles
\[\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

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More answers

anonymous
  • anonymous
I believe so yes, I may have messed up on the first intervals though
freckles
  • freckles
oops all of those a's were suppose to be k's
freckles
  • freckles
because I didn't want to use a for two different things :p
anonymous
  • anonymous
I got you haha, let me see if maybe it would only be on interval on the last two
anonymous
  • anonymous
One*
freckles
  • freckles
like the last one only has two solution so I didn't choose interval style to represent my solutions
anonymous
  • anonymous
This is what I'm encountering
1 Attachment
anonymous
  • anonymous
The last two ended up being only one interval... maybe the first one is only one?
anonymous
  • anonymous
Ah okay, they were all only one interval for some strange reason!
anonymous
  • anonymous
Thanks for your help haha
freckles
  • freckles
one interval for each ?
freckles
  • freckles
the second one I thought should be one interval
freckles
  • freckles
(-4 sqrt(2),4 sqrt(2))
freckles
  • freckles
but the other two look fine to me given your instructions
anonymous
  • anonymous
Yes, so the first one was (4sqrt(2),infinity), second was (0,4sqrt(2)), and third was [4sqrt(2),4sqrt(2)]
anonymous
  • anonymous
For some reason, they just ended up being positive values.
freckles
  • freckles
yeah it looks like they ignored the negative values ...
anonymous
  • anonymous
Yep, well I guess it's easier... weird that they didn't specify that though.
freckles
  • freckles
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
anonymous
  • anonymous
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!
freckles
  • freckles
right
anonymous
  • anonymous
Thanks man, I appreciate your help
freckles
  • freckles
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
anonymous
  • anonymous
haha nice

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