Bessel Differential Equation With Log

[masqau]

I am trying to solved a differential equation of Bessel type,

X^2 Y('')+XY(')+(X^2-n^2)Y+YlogX=0,

where Y(')=d/dx.

Please help me that how to deal with such equation.

 

[Astronuc]

Where did one find such an equation?

A way to solve such an equation is with the Frobenius method.
http://mathworld.wolfram.com/FrobeniusMethod.html

but I'm not sure in this case.

Here is a nice summary of many important differential equations.
http://mathworld.wolfram.com/OrdinaryDifferentialEquation.html

log (x) or ln (x) does not appear in them.

 

[masqau]

Thanks, Astrounus.
actually Frobenius method don't work here because of the singularity of log(x), at x=1.

 

[torquil]

Try to change independent variable from x to u = ln(x). The eq. simplified when I did it, although I haven't double-checked my result.

Torquil

 

[masqau]

Try to change independent variable from x to u = ln(x). The eq. simplified when I did it, although I haven't double-checked my result.

Torquil

Thanks Torquil.
U know the original equation was,
Y''+xy+n^2y+exp(-x)y=0,
when i used the transformation x=exp(-z/2), i got
z^2y''+zy'+(z^2-n^2)y+ylog(z)=0,
which i already post,
now according to u if z=log(u), than i am sure i will get my original equation (1), with exponential.

 

[Astronuc]

Thanks, Astronuc.
actually Frobenius method don't work here because of the singularity of log(x), at x=1.

That was my thinking, but does the equation have to apply at all x, or x=0? I believe the singularity of ln(x) or log(x) is at x=0.

I've sometimes applied a Bessel's equation over a limited range, e.g., finite cylindrical or annular domain.

 

[samreen]

Frobenius ought to work. solve the homogeneous bessels equation to get a complementary soltn usng frob technique, for non integral values of m. for integral vaues, it ought to give linearly dependent solns. and for a particular integral, try variation of parameters. or mayb greens fxn. i haven't started dealing with bessels equation yet, so don't know if dis'l work.

 

[masqau]

I believe the singularity of ln(x) or log(x) is at x=0.

I've sometimes applied a Bessel's equation over a limited range, e.g., finite cylindrical or annular domain.

u r right the singularity is at x=0.
from limit range u mean, to expand the log(x) into to its series form and than take first two or three terms.
u know this problem is very interesting in the sense that it show the clash of linear and exponential behavior.
if the DE is
y''+(n^2+x)y=0, (1)
its solution is Airy function, now if
Y''+(n^2+exp(2))y=0, (2)
its solution is Bessel function. now combining eq.(1) and (2)
y''+(n^2+exp(x)+x)y=0 (3)
eq(3) is the DE which i had posted.

 

[masqau]

Frobenius ought to work. solve the homogeneous bessels equation to get a complementary soltn usng frob technique, for non integral values of m. for integral vaues, it ought to give linearly dependent solns. and for a particular integral, try variation of parameters. or mayb greens fxn. i haven't started dealing with bessels equation yet, so don't know if dis'l work.

Frobenius don't work because of log(x) and its sigularity.
from variation u mean chnage of variable, its also don't work because chnage of variable create image on the other term, like exp(x) have image of log(x) and vice versa.
Green function can be helpful, can u explain it please?

 

[samreen]

sorry. turns out i was kinda wrong about everything else too. greens function is used to seek solutions of inhomogeneous equations. but isn't this one homogeneous? the coefficient of Y (i.e zeroth degree derivative) is (X^2 -m^2 + log X). da log term changes things completely. maybe an ordinary power series expansion abt an ordinary point? the functions regular over the entire finite domain cept at 0 (im pretty sure my suggestions are going from bad to worse)

 

[samreen]

u can preempt the solution, as being possibly of the form f(x)Jn(x), instead of Jn(x). mayb work backwards. I have no idea how the introduction of the logarithmic term is going to change the behaviour of the D.E. bt going by the nature of log fxns, possibly not uniformly.

 

[masqau]

u can preempt the solution, as being possibly of the form f(x)Jn(x), instead of Jn(x). mayb work backwards. I have no idea how the introduction of the logarithmic term is going to change the behaviour of the D.E. bt going by the nature of log fxns, possibly not uniformly.

How about if i put
y=exp(-x)J(x)
hope it works...

 

[samreen]

interesting...lemme try