What is Bessel equation: Definition and 24 Discussions
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions y(x) of Bessel's differential equation
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{\displaystyle x^{2}{\frac {d^{2}y}{dx^{2}}}+x{\frac {dy}{dx}}+\left(x^{2}-\alpha ^{2}\right)y=0}
for an arbitrary complex number α, the order of the Bessel function. Although α and −α produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of α.
The most important cases are when α is an integer or half-integer. Bessel functions for integer α are also known as cylinder functions or the cylindrical harmonics because they appear in the solution to Laplace's equation in cylindrical coordinates. Spherical Bessel functions with half-integer α are obtained when the Helmholtz equation is solved in spherical coordinates.
Hey all,
I wanted to know if anyone knew somewhere I could find the asymptotic behavior for small x (i.e x approaching 0) limit of the modified Bessel equations with complex order. The wikipedia page for Bessel functions...
Hello,
For my homework I am supposed to get-
into the form of a Bessel equation using variable substitution. I am just not sure what substitution to use.
Thanks in advance.
Hello,
i am trying to solve this equation for x
besselj(0,0.5*x)*bessely(0,4.5*x)-besselj(0,4.5*x)*bessely(0,0.5*x) ==0;
I tried vpasolve, but it gave me answer x=0 only. fzero function didnt work, too.
What function can solve this equation?
Thanks
Homework Statement
Homework Equations
The Attempt at a Solution
Because we are only looking at a cross section, I tried to reduce 5.3 down to just being a function of R and Theta. However I reasoned that there should be, based on this problem, no dependence on Theta either, so I figured I...
Homework Statement
Given the bessel equation $$x^2\frac{d^2y}{dx^2} + x\frac{dy}{dx} -(1-x)y=0$$ show that when changing the variable to ##u = 2\sqrt{x}## the equation becomes $$u^2\frac{d^2y}{du^2}+u\frac{dy}{du}+(u^2-4)y = 0$$
Homework Equations
The Attempt at a Solution
##u=2\sqrt{x}##...
Homework Statement
I am trying to find an equation for a free hanging chain of mass m and length L. The chain is hanging vertically downwards where x is measured vertically upwards from the free end of the chain and y is measured horizontally.
Homework Equations
[/B]
I derived this...
Homework Statement
I need to solve a problem like Jackson 3.18. I need to find potential due to the same configuration but the position of two plates is opposite i.e. Plate at Z=0 contains disc with potential V and plate at Z=0 is grounded.
Homework EquationsThe Attempt at a Solution
I think...
I'm trying to show that a function defined with the following recurence relations
$$\frac{dZ_m(x)}{dx}=\frac{1}{2}(Z_{m-1}-Z_{m+1})$$ and $$\frac{2m}{x}Z_m=Z_{m+1}+Z_{m-1}$$ satisfies the Bessel differential equation
$$\frac{d^2}{dx^2}Z_m+\frac{1}{x}\frac{d}{dx}Z_m+(1-\frac{m^2}{x^2})Z_m=0$$...
We first express Bessel's Equation in Sturm-Liouville form through a substitution:
Next, we consider a series solution and replace v by m where m is an integer. We obtain a recurrence relation:
Then, since all these terms must be = 0,
Consider m = 0
First term vanishes and second term =
a1x...
Show that the Legendre equation as well as the Bessel equation for n=integer are Sturm Liouville equations and thus their solutions are orthogonal. How I can proove that ..?
:(
Hi guys, I have this question on Laplace transforms, but am not sure how to start it.
The zero order Bessel function Jo(t) satisfies the ordinary differential equation:
tJ''o(t) + J'o(t) + tJo(t) = 0
Take the Laplace transform of this equation and use the properties
of the transform to find...
Homework Statement
The following is an integral form of the Bessel equation of order n:
J_n(x) = \frac{1}{\pi}\int_0^{\pi}\ \cos(x\sin(t)-nt)\ dt
Show by substitution that this satisfies the Bessel equation of order n.
Homework Equations
Bessel equation of order n: x^2y'' + xy' +...
Hi there. I'm working with the Bessel equation, and I have this problem. It says:
a) Given the equation
\frac{d^2y}{dt^2}+\frac{1}{t}\frac{dy}{dt}+4t^2y(t)=0
Use the substitution x=t^2 to find the general solution
b) Find the particular solution that verifies y(0)=5
c) Does any solution...
Not sure if this is the right place. Mathematica has a function BesselK[0,x] that returns the value of the modified Bessel function K_0 at x. Is there public documentation of how this algorithm works? If not, is there documentation regarding any algorithm of K_0? I am hoping it doesn't...
problem in Bessel equation help ...
Homework Statement
using the formula d\dx (x^n Jn(x))=x^n Jn-1(x)
& 2n\x Jn(x)=Jn+1(x)+Jn-1(x)
Homework Equations
prove that integral from 0 to 1 (x(1-x^2)Jdot(x) dx = 4 J1(1) - 2 Jdot (1)
The Attempt at a Solution
it's difficult one i can not...
Hi all can anyone help me to reduce following diff.Equ. to bessel eq.
4x^3*y''-y=0
thanks in advance .
I am also still trying to show that it can be converted to bessel function.
I remember some of my linear algebra from my studies but can't wrap my head around this one.
Homework Statement
Say my solution to a DE is "f(x)" (happens to be bessel's equation), and it contains a constant variable "d" in the argument of the bessel's functions (i.,e. J(d*x) and Y(d*x)). So...
I need to convertx^{2}y''+2xy'+[kx^{2}-n(n+1)]y=0 using y=x^{-\frac{1}{2}}w to a normal Modified Bessel Equation and I cannot get to that. I check many times and I must be having a blind spot!
This is my work:
y=x^{-\frac{1}{2}} w \Rightarrow...
In the solution to a recent problem set, my prof referenced a "general Bessel ODE" which he gave in the form:
x^{2}\frac{d^{2}y}{dx^{2}}+x\left(a+2bx^{q}\right)\frac{dy}{dx}+\left[c+dx^{2s}-b\left(1-a-q\right)x^{q}+b^{2}x^{2q}\right]y=0
The only format of the Bessel ODE that appears in the...
What am I missing when I'm unsuccessful in showing by direct substitution into the spherical Bessel equation
r^2 \frac{d^2R}{dr^2} + 2r \frac{dR}{dr} + [k^2 r^2 - n(n + 1)] R = 0
that
n_0 (x) = - \frac{1}{x} \sum_{s \geq 0} \frac{(-1)^s}{(2s)!} x^{2s}
is a solution?
What's the catch??