April 20th, 2013, 03:45 AM  #1 
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  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 . 
May 4th, 2013, 12:46 PM  #2 
Senior Member Joined: Feb 2013 Posts: 153 Thanks: 0  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].

May 4th, 2013, 09:59 PM  #3  
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: General Solution of a PDE Quote:
Quote:
 
May 5th, 2013, 02:19 AM  #4 
Senior Member Joined: Feb 2013 Posts: 153 Thanks: 0  Re: General Solution of a PDE
Sorry. Mistake. I forgot f before (xy). So z=a*f(xy)+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.

May 5th, 2013, 03:38 AM  #5  
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: General Solution of a PDE Quote:
Quote:
Quote:
 
May 5th, 2013, 10:37 AM  #6 
Senior Member Joined: Feb 2013 Posts: 153 Thanks: 0  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. 
May 5th, 2013, 10:40 AM  #7  
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: General Solution of a PDE Quote:
 
May 5th, 2013, 10:44 AM  #8 
Senior Member Joined: Feb 2013 Posts: 153 Thanks: 0  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.

May 5th, 2013, 10:50 AM  #9  
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: General Solution of a PDE Quote:
 
May 5th, 2013, 10:54 AM  #10 
Senior Member Joined: Feb 2013 Posts: 153 Thanks: 0  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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