The classic second order differential equation

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If we have [itex]\frac{1}{X(x)}[/itex] [itex]\frac{d^2 X}{dx^2}[/itex]=-κ^2, the literature is saying that the solution must be: e^(±iκx), but am always getting e^(±k^2x).

Isn't the approach is to decently integrate twice and then raise the ln by the exponential? Where am I going wrong? Thanks
 
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arildno
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What do you mean with "integrating twice"?
Show your steps.
 
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jambaugh
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The devil is in the details... could you show your integration work?

Keep in mind that to integrate you need a differential instead of derivative format. letting [itex]Y=\frac{dX}{dx}[/itex] your equation is of the form:
[tex]\frac{1}{X}\frac{dY}{dx} = -k^2[/tex] so
[tex]dY = -k^2 X dx[/tex]
Integrating doesn't work out directly:
[tex] Y(x)=k^2 \int X(x)dx[/tex]
You can try different integration by parts and substitutions... but eventually you come back to the problem of finding an integration factor for the original equation:

[tex] YdY = -k^2 XYdx = -k^2 XdX[/tex]
[tex] Y^2/2 = -k^2 X^2/2[/tex]
[tex]Y = -ik X[/tex]
[tex]dX = -ikX dx, dX/X = -ik dx[/tex]
[tex] X = e^{-ikx}[/tex]
(here I'm sloppy with the integration constants, you need to work them in.)
 
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Jambaugh, thank you for the reply. But that was derivation with second order. You worked it out with first order, right?
 
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One more question, if for x=0, X=0 then X=sinkx according to your result? Or would it be isinkx? Am just confused with the imaginary number..
 
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arildno
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Show your work, and we'll tell you where you are mistaken.
 
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LCKurtz
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If we have [itex]\frac{1}{X(x)}[/itex] [itex]\frac{d^2 X}{dx^2}[/itex]=-κ^2, the literature is saying that the solution must be: e^(±iκx), but am always getting e^(±k^2x).

Isn't the approach is to decently integrate twice and then raise the ln by the exponential? Where am I going wrong? Thanks
It is just a constant coefficient equation ##X''+k^2X=0## with characteristic equation ##r^2+k^2=0##. So ##r=\pm ik##.
 

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