Second Order Differential equation Bessel-type

[Juan Carlos]

Hello!

Im trying to solve this second order differential equation:

\begin{equation*}
-\dfrac{d^2y}{dx^2}+\dfrac{3}{x}\dfrac{dy}{dx}+(x^2+gx^4+2)y=0
\end{equation*}

Any idea?

Maybe it could be converted to a Bessel-like equation (?) with an appropriate change of variables.

The equation arises when your are considering a -2 dimensional (yes!, its correct: "Negative dimension") anhamonic oscillator.

Thanks!

 

[Ssnow]

Hi, there is a proposition that says, if you have an equation of this kind:

$y''+P(x)y'+Q(x)y=0$
where $P$ and $Q$ continuous on $I$ and if $\lambda(x)$ is a particular solution with $\lambda(x)\not=0$, then the general solution is:

$y(x)=\lambda(x)\left(c_{1}+c_{2}\int\frac{e^{-\int P(x)dx}}{\lambda(x)^2}dx\right)$

where $c_{1},c_{2}$ are constants.

So you can start to search a particular solution and then apply this...

 

[the_wolfman]

I not sure how helpful this will be, but I think you're equation is a general confluent hypergeometric differential equation: http://mathworld.wolfram.com/GeneralConfluentHypergeometricDifferentialEquation.html.

The general solutions are:

$y_1 = x^{-A} e^{-m\left(x\right)} F\left(a;b; h\left(x\right)\right)$
$y_2 = x^{-A} e^{-m\left(x\right)} U\left(a;b; h\left(x\right)\right)$

where $F\left(a;b;h\left(x\right)\right)$ and $U\left(a;b;h\left(x\right)\right)$ are confluent hypergeometric functions of the first and second kind.
Admittedly it can be challenging to identify the functions $m\left(x\right)$ and $h\left(x\right)$ . I'd try a few simple functions to see if you can make it work.

Alternatively what is the constant g? Is it a small number? Can you use a perturbative expansion to find an approximate solution?

 

[Juan Carlos]

Hi, there is a proposition that says, if you have an equation of this kind:

$y''+P(x)y'+Q(x)y=0$
where $P$ and $Q$ continuous on $I$ and if $\lambda(x)$ is a particular solution with $\lambda(x)\not=0$, then the general solution is:

$y(x)=\lambda(x)\left(c_{1}+c_{2}\int\frac{e^{-\int P(x)dx}}{\lambda(x)^2}dx\right)$

where $c_{1},c_{2}$ are constants.

So you can start to search a particular solution and then apply this...

Hi, there is a proposition that says, if you have an equation of this kind:

$y''+P(x)y'+Q(x)y=0$
where $P$ and $Q$ continuous on $I$ and if $\lambda(x)$ is a particular solution with $\lambda(x)\not=0$, then the general solution is:

$y(x)=\lambda(x)\left(c_{1}+c_{2}\int\frac{e^{-\int P(x)dx}}{\lambda(x)^2}dx\right)$

where $c_{1},c_{2}$ are constants.

So you can start to search a particular solution and then apply this...

Variation of parameters! I'll give it a try!

Thanks

 

[Juan Carlos]

I not sure how helpful this will be, but I think you're equation is a general confluent hypergeometric differential equation: http://mathworld.wolfram.com/GeneralConfluentHypergeometricDifferentialEquation.html.

The general solutions are:

$y_1 = x^{-A} e^{-m\left(x\right)} F\left(a;b; h\left(x\right)\right)$
$y_2 = x^{-A} e^{-m\left(x\right)} U\left(a;b; h\left(x\right)\right)$

where $F\left(a;b;h\left(x\right)\right)$ and $U\left(a;b;h\left(x\right)\right)$ are confluent hypergeometric functions of the first and second kind.
Admittedly it can be challenging to identify the functions $m\left(x\right)$ and $h\left(x\right)$ . I'd try a few simple functions to see if you can make it work.

Alternatively what is the constant g? Is it a small number? Can you use a perturbative expansion to find an approximate solution?

Thanks I will try with this "factorization" procedure.

 

[ModestyKing]

The Frobenius method will give you solutions I'm sure, they may be series solutions but I don't know what kind of solution you need. http://mathworld.wolfram.com/FrobeniusMethod.html