anonymous
  • anonymous
Can someone help me with this initial value problem? d^2y over dt^2 = 2/t^3. Given is: dr/dt = t=1 and that whole thing = 1. Another Given is: r(1) = 1. I have no idea how to go about this problem after finding the antiderivative for the d^2r over dx^2.
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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schrodinger
  • schrodinger
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freckles
  • freckles
so you never meant to say anything about y right?
anonymous
  • anonymous
Sorry meant r not y.
freckles
  • freckles
\[r''=2t^{-3}\]

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freckles
  • freckles
So you said you integrated both sides: \[r'=2 \cdot \frac{t^{-3+1}}{-3+1}+C_1=2 \cdot \frac{t^{-2}}{-2}+C_1=-1 t^{-2}+C_1\] => \[r'=\frac{-1}{t^2}+C_1\]
freckles
  • freckles
So we are given r'(1)=1?
anonymous
  • anonymous
Just r(1)=1
freckles
  • freckles
dr/dt = t=1? r'=t=1 I look at that is when t=1 ,r '=1
freckles
  • freckles
\[1=\frac{-1}{1^2}+C_1 \] Solve for C_1
anonymous
  • anonymous
where does the r' come from?
anonymous
  • anonymous
Oh! so another way to look at d^2r/dt^2 is r''?
freckles
  • freckles
yep yep
anonymous
  • anonymous
Okay! Also just to clarify, when substituting t and r, just plug in one in newly acquired derivative where r and t are respectively? If r(1)=1 and t(1)=2, then plug in respectively and then ... what would happen?
freckles
  • freckles
well if you are given r'(a)=b then you use the function for r' If t=a then r'=b ------------------------ and if you are given r(c)=d then use the function r If t=c, then r=d You will use both of this conditions to find your constants
anonymous
  • anonymous
So, since r(1)=1, then t(1)=2, then r of t = 1? Somehow I confused myself...
freckles
  • freckles
t(1)=2 makes no sense
freckles
  • freckles
I'm sorry to be so blunt
anonymous
  • anonymous
No, it makes no sense to me either.
freckles
  • freckles
You are given r'(1)=1 and r(1)=1
freckles
  • freckles
|dw:1332451923490:dw|
freckles
  • freckles
r'(1)=1 says when t=1 r'=1
freckles
  • freckles
\[1=\frac{-1}{1^2}+C_1 \] The reason why I wrote this
freckles
  • freckles
I replaced t with 1 and I replaced r' with 1
anonymous
  • anonymous
That makes sense...what happens to the 2 of t(1)=2 tho? how would I solve for C then? Or is it just replace r' with 1 and t with 1?
anonymous
  • anonymous
Sorry for all these questions...
freckles
  • freckles
I don;t understand where you are seeing t(1)=2?
anonymous
  • anonymous
I added it after getting the orignal problem in the "what happens after" part...cause the next problem was basically this problem, but instead of t(1)=1, it was t(2)=1...Sorry if this was all confusing! I don't quite get how to solve for C when the r' and t equal different numbers but have the same number plugged in...
anonymous
  • anonymous
@sharathe you're confused between the input t = 1, output r' ( 1) = 1 From these input and output => constant C1 = 2

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