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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
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so you never meant to say anything about y right?
Sorry meant r not y.
\[r''=2t^{-3}\]

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So you said you integrated both sides: \[r'=2 \cdot \frac{t^{-3+1}}{-3+1}+C_1=2 \cdot \frac{t^{-2}}{-2}+C_1=-1 t^{-2}+C_1\] => \[r'=\frac{-1}{t^2}+C_1\]
So we are given r'(1)=1?
Just r(1)=1
dr/dt = t=1? r'=t=1 I look at that is when t=1 ,r '=1
\[1=\frac{-1}{1^2}+C_1 \] Solve for C_1
where does the r' come from?
Oh! so another way to look at d^2r/dt^2 is r''?
yep yep
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?
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
So, since r(1)=1, then t(1)=2, then r of t = 1? Somehow I confused myself...
t(1)=2 makes no sense
I'm sorry to be so blunt
No, it makes no sense to me either.
You are given r'(1)=1 and r(1)=1
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r'(1)=1 says when t=1 r'=1
\[1=\frac{-1}{1^2}+C_1 \] The reason why I wrote this
I replaced t with 1 and I replaced r' with 1
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?
Sorry for all these questions...
I don;t understand where you are seeing t(1)=2?
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...
@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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