You are Here: Home >< Maths

# Integration question watch

1. To cut it short, I'm doing a question and have arrived at the following differential equations:

r"-r(p')^2=-dv/dr (where the ' indications differentiation w.r.t a variable, say 't')

and p'=c/r^2 where c is a constant.

I am also given that r=asinp.

From all this I have derived the following expressions:

r'=acos(p)p'
r''=-asin(p)(p')^2+acos(p)p''
p''=-2cr'/r^3
r'=cacosp/r^2

The question asks me to show that V is proportional to r^-4.

I can see that this involves seperating variables in the first equation, subbing in those derived expressions and integrating. However, I'm not too sure how to do this- since there are terms like p'' which are related to r.

2. The question asks me to show that V is proportional to r^-4.

I can see that this involves seperating variables in the first equation, subbing in those derived expressions and integrating. However, I'm not too sure how to do this- since there are terms like p'' which are related to r.

do you mean you have this equation to start:

??

If so I would suggest the following:

implies

so

eliminate p from this using the identity

Now sub this expression for dp into the original equation, which eliminates all references to p. I haven't checked but I imagine it can be solved now..?
3. Or do you mean this one

If this, then p isn't in the expression at all..?
do you mean you have this equation to start:

??

If so I would suggest the following:

implies

so

eliminate p from this using the identity

Now sub this expression for dp into the original equation, which eliminates all references to p. I haven't checked but I imagine it can be solved now..?
Thanks, but as far as I can see r is a function of t so it won't really help doing that. This is the main problem I am confused as to how to integrate differentials w.r.t to a variable that isn't the variable which the differential is w.r.t to, if that makes any sense.
5. Thanks, but as far as I can see r is a function of t so it won't really help doing that. This is the main problem I am confused as to how to integrate differentials w.r.t to a variable that isn't the variable which the differential is w.r.t to, if that makes any sense.
yeah I get you, like what is

for example?

At a guess perhaps this

but then I get your problem of what if r is a function of x, i.e. it should also be involved in the integration process. Im afraid im not sure what to do there - might be time to involve a teacher/lecturer/textbook!
yeah I get you, like what is

for example?

At a guess perhaps this

but then I get your problem of what if r is a function of x, i.e. it should also be involved in the integration process. Im afraid im not sure what to do there - might be time to involve a teacher/lecturer/textbook!
Sort of, more like:

7. (Original post by jmz34)
Sort of, more like:

In that case i'd try to find dr/dt explicitly and substitute in the expression I get for it into the integral, but im guessing you tried this already and it isn't possible or something?
In that case i'd try to find dr/dt explicitly and substitute in the expression I get for it into the integral, but im guessing you tried this already and it isn't possible or something?
That integral expression is just a simplified one to show you what I mean, but in the question its the second differential of r w.r.t t, and if you look back I do have an expression for r'' but its in terms of p'' which wouldn't really help since I would need to change dr to acosdp and I'd be in a similar situation.
9. What is v here? (Unless there is some relationship you haven't disclosed, I don't see how v is unique - as you only reference dv/dr, adding a constant to v doesn't affect anything).
10. (Original post by DFranklin)
What is v here? (Unless there is some relationship you haven't disclosed, I don't see how v is unique - as you only reference dv/dr, adding a constant to v doesn't affect anything).
Sorry, in my attempt to keep the question short I've forgotten to mention that. V is a function of r and r only.
11. That integral expression is just a simplified one to show you what I mean, but in the question its the second differential of r w.r.t t, and if you look back I do have an expression for r'' but its in terms of p'' which wouldn't really help since I would need to change dr to acosdp and I'd be in a similar situation.
Ok I get it now.

gives

and

We also have

Therefore

and

So sub this into the two above expressions to get

implies

and

Now the overall expression can be integrated, as you have expressions for dr/dt and d^2r/dt^2 purely in terms of r.

I think that is correct - there may be a flaw i've yet to spot!
12. EDIT - found a flaw already lol.

Thats not right... Ah well I tried!
13. EDIT EDIT

Should be this

So we're back on again...
EDIT EDIT

Should be this

So we're back on again...
Isn't it c^2/r^4?
15. Isn't it c^2/r^4?
Yes it is. lol. Fixed the dt^2 mistake, only to make another...
Yes it is. lol. Fixed the dt^2 mistake, only to make another...
Your expression for the second differential of r w.r.t t is incorrect.
17. Your expression for the second differential of r w.r.t t is incorrect.
yes - should be

Sorry bout that, seem to be all about the mistakes today...
yes - should be

Sorry bout that, seem to be all about the mistakes today...
But surely r'' is zero now?

Also 'spread_logic_not_hate' please feel free to continue to try this, thanks for all the help so far dude.
19. Again this is wrong, r'=(c/r^2)SQRT(a^2-r^2), you have to differentiate this using the product rule to get r''.
Ok now I get

Do you get the same?
20. poop

### Related university courses

TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

This forum is supported by:
Updated: February 3, 2010
Today on TSR

### Top unis in Clearing

Tons of places at all these high-ranking unis

Poll
Useful resources

### Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

### How to use LaTex

Writing equations the easy way

### Study habits of A* students

Top tips from students who have already aced their exams