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#1 2013-01-12 21:06:44
- Fistfiz
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Wronksian determinant in 2nd order linear DE
Hi guys,
I'm studying some demonstrations about 2nd order differential equations of the form:
y''+2by'+ay=f(t)
where a,b are constants.
Suppose that u,v are linearly independent solutions. Now, in several demonstrations, it's needed that the Wronksian determinant of u,v it's different from zero.
I see from Abel's identity (http://en.wikipedia.org/wiki/Abel's_identity) that if this is true for some t0 value, then it's true for all t. Provided this, can I always say that the Wronksian of u,v is always non-zero??
30+2=28 (Mom's identity)
#2 2013-01-12 23:35:41
- scientia
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Re: Wronksian determinant in 2nd order linear DE
Yes, if you can show that for some then for all .
#3 2013-01-13 01:06:52
- Fistfiz
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Re: Wronksian determinant in 2nd order linear DE
Of course 
but what i meant was: can I avoid to include W(t0)!=0 for some t0 in my hypotesis? In other words, if I have two linearly independent solutions u and v, can I automatically say W(u,v)!=0 for all t?
30+2=28 (Mom's identity)
#4 2013-01-13 01:39:21
- scientia
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Re: Wronksian determinant in 2nd order linear DE
I see. Well, if are differentiable, then linear independence implies for all . If they are not both differentiable, then it is possible that they are linearly independent yet . See http://en.wikipedia.org/wiki/Wronskian# … dependence for an example.
