question about neighboorhood at the origin for a solution to a ODE

**Fractalus** · July 18th 2011, 05:25 AM

Find the solution to $x^2y'' + xy' + y = 0$ in terms of power series in the neighbourhood of the origin.

The only real solution that I find is $y = 0$ . Do you know if a solution of the form $y = c_1cos(ln x) + c_2sin(ln x) , x > 0$ is considered to be in the neighbourdhood of 0 ?

**chisigma** · July 18th 2011, 07:32 AM

Originally Posted by Fractalus

Find the solution to $x^2y'' + xy' + y = 0$ in terms of power series in the neighbourhood of the origin.

The only real solution that I find is $y = 0$ . Do you know if a solution of the form $y = c_1cos(ln x) + c_2sin(ln x) , x > 0$ is considered to be in the neighbourdhood of 0 ?

The ODE is 'Euler's type' and its solution is...

$y(x)= c_{1}\ e^{i\ \ln x} + c_{2}\ e^{-i\ \ln x}$ (1)

Neither $e^{i\ \ln x}$ nor $e^{-i\ \ln x}$ is analytic in $x=0$ so that the only 'analytic solution' is for $c_{1}=c_{2}=0$ ...

Kind regards

$\chi$ $\sigma$

**Fractalus** · July 18th 2011, 07:39 AM

Originally Posted by chisigma

The ODE is 'Euler's type' and its solution is...

$y(x)= c_{1}\ e^{i\ x} + c_{2}\ e^{-i\ x}$ (1)

Neither $e^{i\ x}$ nor $e^{-i\ x}$ is analytic in $x=0$ so that the only 'analytic solution' is for $c_{1}=c_{2}=0$ ...

Kind regards

$\chi$ $\sigma$

Thank you for your answer chisigma. I knew this was Euler type but I have to find the solution in terms of $\underline{power series}$ in the neighbourhood of 0.

The solution I gave in my last post is correct for $]0,\infty[$ . But I don't know if it is considered to be in the neighbourhoud of 0.

Do you understand how I got my solution? You have to use the Euler formula.

Is it helping you?

Regards,
Fractalus

**chisigma** · July 18th 2011, 07:55 AM

Originally Posted by Fractalus

Thank you for your answer chisigma. I knew this was Euler type but I have to find the solution in terms of $\underline{power series}$ in the neighbourhood of 0.

The solution I gave in my last post is correct for $]0,\infty[$ . But I don't know if it is considered to be in the neighbourhoud of 0.

Do you understand how I got my solution? You have to use the Euler formula.

Is it helping you?

Regards,
Fractalus

First I must apologize because the errors contained in first stesure of post, errors caused by the 'hurry' for the presence of 'competitors'

...

The 'general solution' of ODE is...

$y(x)= c_{1}\ (\cos \ln x + i\ \sin \ln x) + c_{2}\ (\cos \ln x - i\ \sin \ln x)$ (1)

Now function like $\cos x$ or $\sin x$ are analytic in $x=0$ so that their Taylor series expansion 'somewhere around' $x=0$ exists. But function like $\cos \ln x$ or $\sin \ln x$ aren't analytic around $x=0$ and that is why $\ln x$ doesn't have neither Taylor nor even Laurent expansion around $x=0$ ...

Kind regards

$\chi$ $\sigma$

**FernandoRevilla** · July 18th 2011, 08:24 AM

Originally Posted by Fractalus

The solution I gave in my last post is correct for $]0,\infty[$ . But I don't know if it is considered to be in the neighbourhoud of 0.

As $x=0$ is a regular singular point of $y''+y'/x+y/x^2=0$ then, by a well known theorem there exists a solution valid in $0<x<R\leq +\infty$ . The name for $(0,R)$ it is irrelevant. For example "right punctured neighborhood" of $0$ ?

**Fractalus** · July 18th 2011, 08:44 AM

Originally Posted by FernandoRevilla

As $x=0$ is a regular singular point of $y''+y'/x+y/x^2=0$ then, by a well known theorem there exists a solution valid in $0<x<R\leq +\infty$ . The name for $(0,R)$ it is irrelevant. For example "right punctured neighborhood" of $0$ ?

I definetely agree with you. That's what I thought and I wondered if we had to find a neighborhood where 0 is included. So thank you, now I know that my answer is correct.

**chisigma** · July 18th 2011, 09:24 AM

An 'old wolf' may lose his teeth and also his memory so that I remember only now a problem I analysed two years ago. The 'general solution' of the ODE...

$x^{2}\ y^{''} + x\ y^{'} + y = 0$ (1)

... is...

$y(x)= c_{1}\ \text{icos} (x) + c_{2}\ \text{isin} (x)$ (2)

... where icos(*) and isin(*) are what I called 'I-functions' and their properties I briefly described in ...

http://digilander.libero.it/luposaba...-functions.pdf

In particular the Laurent expansion's coefficients of the I-functions around $s=0$ are computed...

Kind regards

$\chi$ $\sigma$

**Fractalus** · July 18th 2011, 09:27 AM

Originally Posted by chisigma

An 'old wolf' may lose his teeth and also his memory so that I remember only now a problem I analysed two years ago. The 'general solution' of the ODE...

$x^{2}\ y^{''} + x\ y^{'} + y = 0$ (1)

... is...

$y(x)= c_{1}\ \text{icos} (x) + c_{2}\ \text{isin} (x)$ (2)

... where icos(*) and isin(*) are what I called 'I-functions' and theis properties I briefly described in ...

http://digilander.libero.it/luposaba...-functions.pdf

In particular the Laurent expansion's coefficients of the I'functions around $s=0$ are computed...

Kind regards

$\chi$ $\sigma$

Thanks anyway chisigma, but I think other answer I got is enought. What you gave me is overkill because I'm not doing a complex functions course.

**Fractalus** · July 18th 2011, 05:51 PM

Originally Posted by chisigma

An 'old wolf' may lose his teeth and also his memory so that I remember only now a problem I analysed two years ago. The 'general solution' of the ODE...

$x^{2}\ y^{''} + x\ y^{'} + y = 0$ (1)

... is...

$y(x)= c_{1}\ \text{icos} (x) + c_{2}\ \text{isin} (x)$ (2)

... where icos(*) and isin(*) are what I called 'I-functions' and their properties I briefly described in ...

http://digilander.libero.it/luposaba...-functions.pdf

In particular the Laurent expansion's coefficients of the I-functions around $s=0$ are computed...

Kind regards

$\chi$ $\sigma$

But your article is still very interesting!

**chisigma** · July 19th 2011, 01:15 AM

Originally Posted by chisigma

An 'old wolf' may lose his teeth and also his memory so that I remember only now a problem I analysed two years ago. The 'general solution' of the ODE...

$x^{2}\ y^{''} + x\ y^{'} + y = 0$ (1)

... is...

$y(x)= c_{1}\ \text{icos} (x) + c_{2}\ \text{isin} (x)$ (2)

... where icos(*) and isin(*) are what I called 'I-functions' and their properties I briefly described in ...

http://digilander.libero.it/luposaba...-functions.pdf

In particular the Laurent expansion's coefficients of the I-functions around $s=0$ are computed...

Kind regards

$\chi$ $\sigma$

The paper about the 'I-functions' written two years ago had never been controlled... today I discovered some errors [I apologize for that

...] and they have been corrected...

Kind regards

$\chi$ $\sigma$

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