Math Is Fun Forum

lindah

Member

Registered: 2010-01-25

Posts: 121

Separation of variables: DE

Hi guys,
I am having a bit of difficulty obtaining an answer for this, since I suspect the answer will be in implicit form.
May I ask if I have set up the separation of variables correctly?

First I have:

Substituting these into original equation:

From here I integrate both sides but obtain:

I am not sure how to express this in terms of w to obtain the solution to y(x)

Any tips would be greatly appreciated!!!!
tia
Linda

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Separation of variables: DE

Hi lindah;

I am getting:

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Separation of variables: DE

hi Lindah,

I got

Not sure why mine lacks a "-3" but the "c" will take care of it anyway.

The question, as posted, doesn't say you have to make "y2 the subject of the expression.  Maybe what you have is sufficient ??

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Separation of variables: DE

Hold on.  Make that:

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

lindah

Member

Registered: 2010-01-25

Posts: 121

Re: Separation of variables: DE

Hi bob and bobbym;

When I do the calculations by hand I get (2w-4), but when I run it through Mathematica I get (4-2w)
I am not too sure why the difference

Yes the question doesn't seem to ask for that. When I asked my lecturer he was very vague.
In your experiences, have you guys ever had to go further than that answer?

Thank you for taking a look!!!
Linda

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Separation of variables: DE

You could substitute back for y and x  and then solve for y. You will get an answer in terms of the Lambert function.

Hold it Bob changed. I would go with Mathematica's answer.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Separation of variables: DE

The question says "solve" and it's in terms of x and y.

So put "y" back in and then, as it doesn't ask for "y = ...." ,   I think you can stop.

Still thinking about the (2w - 4)      (4 - 2w) issue.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Separation of variables: DE

The log of a minus is often the same as the inverse plus.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

lindah

Member

Registered: 2010-01-25

Posts: 121

Re: Separation of variables: DE

Hi bob and bobbym;

Thank you for the feedback!!!
I will follow up with my lecturer (its an assignment question) since we haven't covered Lambert/Product Logs before, though I can see Mathematica expresses it in that form.

As I need to show working how I obtain my answer, I just tried the integration by hand again and see no opportunity (yet) for it to become (4-2w)

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Separation of variables: DE

edit:  No. Nor did I.

Back to first principles.  (apologies if what follows is obvious)

Suppose

but

and

So far, so good.

But ..... ln(x) is only defined in reals for x > 0

So it is usual to put

Now, I'm not sure how that effects a problem where x < 0.

As y = 1/x has rotational symmetry around (0,0) it is probably ok.

So it seems to be arbitary whether you write ln|2w-4| or ln|4-2w|.

They seem to mean the same thing.

I assume that buried deep in the routines that mathematica uses, it has a line of code that says write the latter rather than the former. 

Bob

Last edited by Bob (2011-09-22 01:20:15)

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Separation of variables: DE

This may be the reason.  log( 1/ 2) = - log(2)

Yes, I think it does not matter.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Separation of variables: DE

hi

I  don't think it's the sign before the log that's in question.  My post #3 was wrong.  In post #2, #4 and #5 we all have a plus.  The question is about what it's the log of.

I think log|2w-4| = log|4-2w|    so either answer is ok.

Got to get back to the show preparations now.  I'll look in again later. 

Bob

Last edited by Bob (2011-09-22 01:28:22)

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Separation of variables: DE

Agreed, Bob.

lindah:

What happened to the Stochastic Calculus?

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

lindah

Member

Registered: 2010-01-25

Posts: 121

Re: Separation of variables: DE

Hi guys,

It makes sense to me when taking logarithms it is the absolute value.

Thank you very much for clearer that up!!!
Linda

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Separation of variables: DE

Hi lindah;

You are welcome, I never got around to saying thanks for telling me about the hike you took in Tasmania, I think. You sent me to a page explaining it, I had no idea there were so many rules to be followed.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

lindah

Member

Registered: 2010-01-25

Posts: 121

Re: Separation of variables: DE

What happened to the Stochastic Calculus?

Hi bobbym;

I had to reorganize my course progression so that I'm enrolled in 4 subjects per semester.
If I had taken up Stochastic Calculus this semester, I would only have 3 subjects available next semester.
This subject (DE) won't be available until Spring next year, while Stochastic Calculus is available all year around.

I was initially excited to do Stochastic Calculus, but upon looking at some texts, I am slightly glad it won't be until next year

lindah

Member

Registered: 2010-01-25

Posts: 121

Re: Separation of variables: DE

Hi lindah;

You are welcome, I never got around to saying thanks for telling me about the hike you took in Tasmania, I think. You sent me to a page explaining it, I had no idea there were so many rules to be followed.

Hi bobby;

Yes, you did say thanks a while back!
Some parts are World Heritage Listed and there have been lots of cases of people getting lost and injured since some hikes can be catered to being only biking and kayaking, so they are quite stringent with the rules.

Are you a bit of hiker?

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Separation of variables: DE

Not since Lake Tahoe, and that was not all that much.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

lindah

Member

Registered: 2010-01-25

Posts: 121

Re: Separation of variables: DE

Alternative working for this question as I am writing up my final atm

I'm curious as to why ln|w-2| also works, and this time around, instead of using integration by parts, I also do not have the +3 like bob initially.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Separation of variables: DE

Hi;

That working is correct. In the other solution a ln(2) can be pulled out and absorbed into the constant.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

lindah

Member

Registered: 2010-01-25

Posts: 121

Re: Separation of variables: DE

Hi bobbym;

Sorry about that!
I'm having my blonde moments (its late here) and dreaming of nothing but differential equations atm

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Separation of variables: DE

Well now, that is scary! Dreaming of DE's!

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.