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-   -   (y')^2+y^2=-2 why this equation has no general solution ? (http://www.physicsforums.com/showthread.php?t=438409)

young_eng Oct15-10 05:41 PM

(y')^2+y^2=-2 why this equation has no general solution ?
 
hi

(y')^2+y^2=-2
why this differential equation has no general solution ?

Pengwuino Oct15-10 07:01 PM

Re: (y')^2+y^2=-2 why this equation has no general solution ?
 
It's non-linear. Having a general solution for these types of equations is the exception, not the rule.

Not that there isn't a general solution because I don't know. It's just that non-linear equations rarely have general solutions

Dickfore Oct15-10 07:19 PM

Re: (y')^2+y^2=-2 why this equation has no general solution ?
 
Because the lhs is necessarily non-negative (sum of squares), whereas the lhs is negative. You can have a solution in complex numbers.

jackmell Oct16-10 12:14 AM

Re: (y')^2+y^2=-2 why this equation has no general solution ?
 
Quote:

Quote by young_eng (Post 2935154)
hi

(y')^2+y^2=-2
why this differential equation has no general solution ?

Why not just solve it the regular way:

[tex]\frac{dy}{\sqrt{-2-y^2}}=\pm dx[/tex]

or [itex]y=\pm i\sqrt{2}[/tex] are solutions, maybe singular ones. Not sure. Otherwise:

[tex]\frac{y\sqrt{-2-y^2}}{2+y^2}=\tan(c\pm x)[/tex]

[tex]y(x)=\pm \frac{\sqrt{2}\tan(c\pm x)}{\sqrt{-\sec^2(c\pm x)}}[/tex]

so that the solution is in the form of y(z)=u+iv

HallsofIvy Oct17-10 11:31 AM

Re: (y')^2+y^2=-2 why this equation has no general solution ?
 
There is no general solution in terms of real valued functions because if y' and y are both real numbers (for a given x) then [itex](y')^2+ y^2[/itex] cannot be negative!

Oops! Dickfore had already said that, hadn't he?


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