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April 20th, 2013, 03:45 AM   #1
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General Solution of a PDE

Consider the following PDE :



I have manged to find out that satisfies the above when is continues and differentiable everywhere on the xy plane.

Is it possible to show that there are no more class of solutions to this one? Even if the RHS is is different in the given PDE, it seems AFAIHO, that the solution is dependent on this particular form.

Thanks in advance,
Balarka
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May 4th, 2013, 12:46 PM   #2
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Re: General Solution of a PDE

That is general solution! But, don't forget constant function. For real, this is general solution [attachment=0:261e79zq]PrtScr capture.jpg[/attachment:261e79zq].
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May 4th, 2013, 09:59 PM   #3
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Re: General Solution of a PDE

Quote:
Originally Posted by limes5
For real, this is general solution : . . .
Absolutely not. It is a special case since another solution can't be deduced from that. The general solution is just what I said, where F is differentiable anywhere.

Quote:
Originally Posted by limes5
Don't forget constant function
That is a special case too. Define F as a constant function and you'll get
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May 5th, 2013, 02:19 AM   #4
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Re: General Solution of a PDE

Sorry. Mistake. I forgot f before (x-y). So z=a*f(x-y)+b, that's general solution. For real, it's something that we called "total (complete) integral". From total integral we can get any other integral (solution). If any other solution exist it can be derived using this solution.
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May 5th, 2013, 03:38 AM   #5
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Re: General Solution of a PDE

Quote:
Originally Posted by limes5
z=a*f(x-y)+b, that's general solution.
Which is the same as since in which everything is in terms of x - y.

Quote:
Originally Posted by limes5
For real, it's something that we called "total (complete) integral".
I know, but I am not asking for the name. I want justification that this is the only total integral of the PDE. How could you guarantee that there are no other class of function for which another disjoint set of solution can generated; how can it be mathematically attacked ? Understood the question?

Quote:
Originally Posted by limes5
If any other solution exist it can be derived using this solution.
I don't think this is very obvious, if you have a proof, please post it!
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May 5th, 2013, 10:37 AM   #6
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Re: General Solution of a PDE

I have understood what you tryed to say. When I learned PDE there wasn't any proof of that statement in my book. But i have another good book and I will see if there is any proof. But I haven't that book now with me, and I can do it in few days... :/

Note that a, b are real numbers... I'm not sure you can write as you wrote in previous message.
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May 5th, 2013, 10:40 AM   #7
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Re: General Solution of a PDE

Quote:
Originally Posted by limes5
When I learned PDE there wasn't any proof of that statement in my book.
You have a book that mentions the PDE ? Would you care to mention the name?
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May 5th, 2013, 10:44 AM   #8
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Re: General Solution of a PDE

Not that equation. I'm talking about total (complete) integral. There is a proof of "If any other solution exist it can be derived using this solution" in that book, i think.
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May 5th, 2013, 10:50 AM   #9
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Re: General Solution of a PDE

Quote:
Originally Posted by limes5
If any other solution exist it can be derived using this solution
That's the definition for the total integrals no?
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May 5th, 2013, 10:54 AM   #10
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Re: General Solution of a PDE

[/quote] I don't think this is very obvious, if you have a proof, please post it![/quote]
You ask me to proove the deffinition!
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