## anonymous 5 years ago can anyone help me get started on this? dh/dt= 0.016(2.5-h) Solve the differential equation to find h in terms of t.

1. anonymous

dh/(2.5-h)=0.016 dt now integrating, -log(2.5-h)=0.016 t+c

2. anonymous

Is this enough, ersp?

3. anonymous

i think so my brains kinda died doing this work XD thanks :)

4. anonymous

ok :)

5. anonymous

actually i dont get where the minus log comes from when integrating

6. anonymous

Comes from the -h...

7. anonymous

$\int\limits_{}{}\frac{dh}{2.5-h}=-\int\limits_{}{}\frac{(2.5-h)'dh}{2.5-h}=-\log (2.5-h)+c$

8. anonymous

I got bumped. Is this explanation okay?

9. anonymous

i think so cheers

10. anonymous

so what method is this?

11. anonymous

Method? To solve the differential equation?

12. anonymous

The d.e. is separable.

13. anonymous

Or, you can just take the reciprocal of both sides and integrate directly, but it's the same thing.

14. anonymous

thanks makes a bit more sense to me now, my mind is really failing me today!

15. anonymous

np ;)

16. anonymous

shouldnt both sides be logged?

17. anonymous

No. It goes like this:$\frac{dh}{dt}=0.016(2.5-h) \rightarrow \frac{dh}{2.5-h}=0.016dt \rightarrow \int\limits_{}{}\frac{dh}{2.5-h}=\int\limits_{}{}0.016dt$$\rightarrow -\log (2.5-h) = 0.016t + c \rightarrow \log (2.5-h)=-0.016t+c_1 \rightarrow 2.5-h=c_2e^{-0.016t}$$\rightarrow h=2.5-ce^{-0.016t}$

18. anonymous

I added subscripts as I went along, but we just lump constants into constants, so normally, the subscripts are left off...which is what I did in the end.

19. anonymous

ah that makes sense thanks very much, this work ive been throwing my head in to a wall for the last few days!

20. anonymous

lol, it happens :p

21. anonymous

Thanks for fanning me, ersp :D

22. anonymous

after that much help it seems logical! XD

23. anonymous

ah it doesnt seem to fit for the next part of the question, at max h is 2.5, how long does it take to get to this point, so i have to rearrange it to make t the subject but the values dont seem to fit at all!