## sharathe 3 years ago 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.

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1. freckles

so you never meant to say anything about y right?

2. sharathe

Sorry meant r not y.

3. freckles

$r''=2t^{-3}$

4. freckles

So you said you integrated both sides: $r'=2 \cdot \frac{t^{-3+1}}{-3+1}+C_1=2 \cdot \frac{t^{-2}}{-2}+C_1=-1 t^{-2}+C_1$ => $r'=\frac{-1}{t^2}+C_1$

5. freckles

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

6. sharathe

Just r(1)=1

7. freckles

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

8. freckles

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

9. sharathe

where does the r' come from?

10. sharathe

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

11. freckles

yep yep

12. sharathe

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?

13. 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

14. sharathe

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

15. freckles

t(1)=2 makes no sense

16. freckles

I'm sorry to be so blunt

17. sharathe

No, it makes no sense to me either.

18. freckles

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

19. freckles

|dw:1332451923490:dw|

20. freckles

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

21. freckles

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

22. freckles

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

23. sharathe

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?

24. sharathe

Sorry for all these questions...

25. freckles

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

26. sharathe

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...

27. Chlorophyll

@sharathe you're confused between the input t = 1, output r' ( 1) = 1 From these input and output => constant C1 = 2