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**Fistfiz****Member**- Registered: 2012-07-20
- Posts: 33

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)

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**scientia****Banned**- Registered: 2009-11-13
- Posts: 224

Yes, if you can show that for some then for all .

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**Fistfiz****Member**- Registered: 2012-07-20
- Posts: 33

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)

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**scientia****Banned**- Registered: 2009-11-13
- Posts: 224

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.

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**Fistfiz****Member**- Registered: 2012-07-20
- Posts: 33

Thank you, this remark:

"Peano (1889) observed that if the functions are analytic, then the vanishing of the Wronskian in an interval implies that they are linearly dependent."

opened my eyes

30+2=28 (Mom's identity)

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