Hello Mathematicians :)

I'm currently investigating the differential equation

$$f''=\left(\sum a_n x^n\right)f$$

I was wondering if anyone could please provide me with a lead as to where I may find the Green's functions? Sadly I have not been able to achieve the result through my own computations and an internet / handbook / library search has turned up nothing which I have been able to use

Thank you

Hello

I'm currently investigating the differential equation

$$f''=\left(\sum a_n x^n\right)f$$

I was wondering if anyone could please provide me with a lead as to where I may find the Green's functions? Sadly I have not been able to achieve the result through my own computations

Hello Mathematicians :)

I'm currently investigating the differential equation

$$-\phi''+\left(x^{10}+gx^{4}+\frac{l(l+1)}{x^2}\right)\phi=0$$

which is solvable by

$$\phi(x,l)=x^{l+1}\exp\left(-\frac{x^6}{6}\right){}_1F_1\left(\frac{g}{12}+\frac{l}{6}+\frac{7}{12};\frac{l}{3}+\frac{7}{6};\frac{x^6}{3}\right)$$

and also under the exchange $l\rightarrow - l-1$.

I was wondering if anyone could please provide me with a lead as to where I may find the Green's functions of $\phi(x,l)$ and $\phi(x,-l-1)$? Sadly I have not been able to achieve the result through my own computations and an internet / handbook / library search has turned up nothing which I have been able to use

I would be forever indebted to anyone who could provide a reference where I may find these identities

Many thanks :)

Hello Mathematicians :)

I'm currently investigating the differential equation

$$f''=\left(\sum a_n x^n\right)f$$

I was wondering if anyone could please provide me with a lead as to where I may find the Green's functions? Sadly I have not been able to achieve the result through my own computations and an internet / handbook / library search has turned up nothing which I have been able to use

Thank you

Hello Mathematicians :)

I'm currently investigating the differential equation

$$-\phi''+\left(x^{10}+gx^4+\frac{l(l+1)}{x^2}\right)\phi=0$$

which is solvable by

$$\phi(x,l)=x^{l+1}\exp\left(-2\frac{x^{M/2+1}}{M+2}\right){}_1F_1\left(\frac{2g+4l+4+M}{2M+4};\frac{M+4l+4}{M+2};4\frac{x^{M/2+1}}{M+2}\right)$$

and also under the exchange $l\rightarrow - l-1$.

I was wondering if anyone could please provide me with a lead as to where I may find the Green's functions of $\phi(x,l)$ and $\phi(x,-l-1)$? Sadly I have not been able to achieve the result through my own computations and an internet / handbook / library search has turned up nothing which I have been able to use

I would be forever indebted to anyone who could provide a reference where I may find these identities

Many thanks :)

Hello Mathematicians :)

I'm currently investigating the differential equation

$$-\phi''+\left(x^{10}+gx^{4}+\frac{l(l+1)}{x^2}\right)\phi=0$$

which is solvable by

$$\phi(x,l)=x^{l+1}\exp\left(-\frac{x^6}{6}\right){}_1F_1\left(\frac{g}{12}+\frac{l}{6}+\frac{7}{12};\frac{l}{3}+\frac{7}{6};\frac{x^6}{3}\right)$$

and also under the exchange $l\rightarrow - l-1$.

I was wondering if anyone could please provide me with a lead as to where I may find the Green's functions of $\phi(x,l)$ and $\phi(x,-l-1)$? Sadly I have not been able to achieve the result through my own computations and an internet / handbook / library search has turned up nothing which I have been able to use

I would be forever indebted to anyone who could provide a reference where I may find these identities

Many thanks :)