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Interesting question

i found this question on the internet.
could someone help me with this ?
thanks in advance.

Let

P(t)

be a function from

R

C

.
We know that

Unparseable latex formula:

[br]\dfrac{d}{dt}|\dfrac{dP}{dt}|=0 \\[br]\dfrac{1}{T} \int_0^T Re(P(t))dt = \dfrac{\sqrt{40}-5}{6} \\[br]\dfrac{1}{T} \int_0^T t Im(P(t))dt = \dfrac{\sqrt{5}-\sqrt{8}}{6}[br]

Sketch

P(t)

Reply 1

LightBlueSoldier

What are your thoughts?

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Reply 2

paulk94

well write

P(t)=x(t)+iy(t)

we know that

Unparseable latex formula:

[br]\dfrac{d}{dt}|\dfrac{dP}{dt}|=0 \\[br]

differentiate P(t) two times we come to the conclusion

x'x''=y'y''

and then somehow..find P(t)

i'm not sure if this helps..

Reply 3

LightBlueSoldier

Original post by paulk94

well write

P(t)=x(t)+iy(t)

we know that

Unparseable latex formula:

[br]\dfrac{d}{dt}|\dfrac{dP}{dt}|=0 \\[br]

differentiate P(t) two times we come to the conclusion

x'x''=y'y''

and then somehow..find P(t)

i'm not sure if this helps..

I don't think the last bit is accurate. But anyway unpack the integrals too. Gather information. See if you can spot a trivial solution etc.

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Reply 4

paulk94

Original post by LightBlueSoldier

I don't think the last bit is accurate. But anyway unpack the integrals too. Gather information. See if you can spot a trivial solution etc.

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do you know the answer..?

Reply 5

LightBlueSoldier

Original post by paulk94

do you know the answer..?

The question as loosely defined in the OP has many solutions. I'm sure I could work them out pretty quickly. But that's not the point, the point is for you to work out the answer.

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Reply 6

cac2008

Original post by paulk94

do you know the answer..?

As LBS said, try to spot a trivial case where it all relates. That usually gives an insight as to its properties, which tells you where to go next.

Reply 7

paulk94

Original post by cac2008

As LBS said, try to spot a trivial case where it all relates. That usually gives an insight as to its properties, which tells you where to go next.

well trivial cases:

Unparseable latex formula:

[br]x=\dfrac{\sqrt{10}-5}{6}\\[br]y=\dfrac{\sqrt{5}-\sqrt{8}}{6t}[br]

in this case x,y satisfy the integration conditions.
but it woudn't satisty the first condition...

:confused:

i'm not sure how you guys got to a trivial solution.
few more hints would be great!! thanks!

Reply 8

DFranklin

I'm unconvinced there's any solution to the integral part of the equation other than the one you've found (if you multiply the integral equations by T and then differentiate w.r.t. T it basically forces what x(t) and y(t) must be).

And then as you say it doesn't satisfy the |dP/dt| constraint. :confused:

(edited 9 years ago)

Reply 9

LightBlueSoldier

Original post by paulk94

well trivial cases:

Unparseable latex formula:

[br]x=\dfrac{\sqrt{10}-5}{6}\\[br]y=\dfrac{\sqrt{5}-\sqrt{8}}{6t}[br]

in this case x,y satisfy the integration conditions.
but it woudn't satisty the first condition...

:confused:

i'm not sure how you guys got to a trivial solution.
few more hints would be great!! thanks!

Well if T is a constant then just treat it like one and use the integrals to find a constant. So the function can be a determined constant which clearly satisfies the D.E. Under this treatment you would think that there would be lots of solutions, although it's no clear how you would explicitly find them although it is not impossible to visualize what they must look like. Have you copied down every part of the question?

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Original post by DFranklin