Wikipedia:Reference desk/Archives/Mathematics/2010 June 23

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June 23

hint needed for apparently difficult problem

Hello all... I have a problem which I have been grappling with for some time. Let b be a positive integer and consider the equation z = x + y + b where x,y,z are variables. Suppose the integers {1,2,...4b+5} are partitioned in two classes arbitrarily. I wish to show that at least one of the classes always contains a solution to the equation.

I have tried using induction on b. The case b = 1 has been solved entirely by me. But I cannot understand how to use the induction hypothesis to prove the result. The more I think of it, the more I feel that a different approach to the problem is needed, but I cant figure out what. It is sort of a special case of a research problem, which has been solved in a more general way. I have little experience of doing research on my own, and so will be glad if anyone can offer me any advice or hints. Thanks - Shahab (talk) 07:06, 23 June 2010 (UTC)

Do x, y and z need to be different? -- Meni Rosenfeld (talk) 07:26, 23 June 2010 (UTC)

No.-Shahab (talk) 07:34, 23 June 2010 (UTC)

It looks like a pigeonhole principle problem to me but I haven't figured it out yet. Dmcq (talk) 08:33, 23 June 2010 (UTC)

The other thing I can see is that if you consider x+x+b you tend to need to have the 0 mod 3 and 1 mod 3 ones into separate sets and then the 2 mod 3 ones have to be stuck in and cause problems. Dmcq (talk) 09:48, 23 June 2010 (UTC)

Here is an outline of a solution. Let's call the two partitions A and B and let's try to split the integers from 1 to 4b+5 between A and B so that neither A nor B contains a solution to z=x+y+b.

Let's put 1 in A. Then if b+2 is also in A we have x=1, y=1 and z=x+y+b all in A. So we must put b+2 in B.

Now, if 3b+4 is also in B, we have x=b+2, y=b+2 and z=x+y+b all in B. So we must put 3b+4 in A.

Now we have 1 and 3b+4 both in A. If 4b+5 is in A then we have ... what ? Or if 2b+3 is in A then we have ... what ? And if we put 2b+3 and 4b+5 both in B, along with b+2 which is already in B, then we have ... what ?

(There may be a more elegant solution that uses induction on b and/or the pigeonhole principle, but I can't see it.) Gandalf61 (talk) 14:45, 24 June 2010 (UTC)

That seems plenty elegant to me, thanks, I wish I'd spent a bit more time on it and got something like that. Dmcq (talk) 21:22, 24 June 2010 (UTC)

dy

In one of my textbooks it says $\int dy=[function]dx$ What is meant by the $\int dy$ —Preceding unsigned comment added by 76.230.251.114 (talk • contribs) 17:01, 23 June 2010

Generally the notation $\int dy$ means the same thing as $\int 1\cdot dy$. --Trovatore (talk) 17:50, 23 June 2010 (UTC)

I bet your textbook says $\int dy=\int f(x)dx$ meaning $y=\int f(x)dx$. Bo Jacoby (talk) 19:33, 23 June 2010 (UTC).

Solving equations like this

How do you solve equations like this for y:
${\frac {dy}{dx}}=xy$
Where the derivative of y with respect to x is equated to some function of y and x? —Preceding unsigned comment added by 203.22.23.9 (talk) 22:55, 23 June 2010 (UTC)

See separation of variables. --Tango (talk) 23:00, 23 June 2010 (UTC)

I'm not sure how rigorous this is, but we can multiply both sides by dx and then divide both sides by y. This gives a dx on one side and a dy on the other side, so we integrate to give

$\int {\frac {1}{y}}\,{\mbox{d}}y=\int \!x\,{\mbox{d}}x$

Evaluating these two indefinite integrals gives ln(y) = ½x² + c where c is the constant of integration. We can solve for y by taking the exponentiation of both sides to the base e. Writing k for e^c to simplify things we have

$y=e^{{\frac {1}{2}}x^{2}+c}=ke^{{\frac {1}{2}}x^{2}}.$

We can check this solution, and show that it is right:

${\frac {{\mbox{d}}y}{{\mbox{d}}x}}=kxe^{{\frac {1}{2}}x^{2}},\ \ xy=x\left(ke^{{\frac {1}{2}}x^{2}}\right)=kxe^{{\frac {1}{2}}x^{2}}.$ •• Fly by Night (talk) 13:48, 24 June 2010 (UTC)

From what I remember, the method that is normally taught is the integrating factor method. Rewrite the equation as

${\frac {{\mbox{d}}y}{{\mbox{d}}x}}-xy=0$

Now, our integrating factor is $e^{\int -x\,dx}=e^{-{\frac {1}{2}}x^{2}}$. Multiply through and we get

$e^{-{\frac {1}{2}}x^{2}}{\frac {{\mbox{d}}y}{{\mbox{d}}x}}-e^{-{\frac {1}{2}}x^{2}}xy=0$

This is actually the derivative of a product. Simplifying, we get

${\frac {\mbox{d}}{{\mbox{d}}x}}\left(ye^{-{\frac {1}{2}}x^{2}}\right)=0$

This can be easily integrated and rearranged to get

$y=ce^{{\frac {1}{2}}x^{2}},\,c\in \mathbb {R}$

Readro (talk) 14:16, 24 June 2010 (UTC)

See also separation of variables. 75.57.243.88 (talk) 21:45, 24 June 2010 (UTC)

It's not "also", because Tango already mentioned it. -- Meni Rosenfeld (talk) 15:02, 25 June 2010 (UTC)