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.

- anonymous

- katieb

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- freckles

so you never meant to say anything about y right?

- anonymous

Sorry meant r not y.

- freckles

\[r''=2t^{-3}\]

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## More answers

- 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

So we are given r'(1)=1?

- anonymous

Just r(1)=1

- freckles

dr/dt = t=1?
r'=t=1
I look at that is when t=1 ,r '=1

- freckles

\[1=\frac{-1}{1^2}+C_1 \]
Solve for C_1

- anonymous

where does the r' come from?

- anonymous

Oh! so another way to look at d^2r/dt^2 is r''?

- freckles

yep yep

- 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

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

So, since r(1)=1, then t(1)=2, then r of t = 1? Somehow I confused myself...

- freckles

t(1)=2 makes no sense

- freckles

I'm sorry to be so blunt

- anonymous

No, it makes no sense to me either.

- freckles

You are given r'(1)=1 and r(1)=1

- freckles

|dw:1332451923490:dw|

- freckles

r'(1)=1 says when t=1 r'=1

- freckles

\[1=\frac{-1}{1^2}+C_1 \]
The reason why I wrote this

- freckles

I replaced t with 1 and I replaced r' with 1

- 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

Sorry for all these questions...

- freckles

I don;t understand where you are seeing t(1)=2?

- 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

@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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