Solution of Parametric Bessel's equation.

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In summary, the conversation discusses solving a problem involving the equation x^{2}y''+xy'+(\lambda^{2}x^{2}-\frac{1}{4})=0 with the boundary conditions y(0)=finite and y(\pi)=0. The solution involves using t=\lambda x and finding two linearly independent solutions: J_{\frac{1}{2}}[\sqrt{\frac{2}{\pi\lambda x}}sin(\lambda x)] and J_{-\frac{1}{2}}[\sqrt{\frac{2}{\pi\lambda x}}sin(\lambda x)]. The general solution of y is then given as y=C_{1}J_{\frac{1}{2}}
  • #1
yungman
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Problem to solve: [tex]x^{2}y''+xy'+(\lambda^{2}x^{2}-\frac{1}{4})=0[/tex] subject to [tex]y(0)=finite,y(\pi)=0[/tex]
Let [tex]t=\lambda x[/tex] which give [tex]t^{2}\frac{d^{2}y}{dt^{2}}+t\frac{dy}{dt}+(t^{2}-\frac{1}{2})y=0[/tex]

With that first linear independent solution is [tex]J_{\frac{1}{2}}[\sqrt{\frac{2}{\pi\lambda x}}sin(\lambda x)][/tex] and second independent solution is [tex]J_{-\frac{1}{2}}[\sqrt{\frac{2}{\pi\lambda x}}sin(\lambda x)][/tex]

Since these two are linear independent solution, I should be able to write the general solution of y:

[tex]y=C_{1}J_{\frac{1}{2}}[\sqrt{\frac{2}{\pi\lambda x}}sin(\lambda x)] + C_{2}J_{-\frac{1}{2}}[\sqrt{\frac{2}{\pi\lambda x}}sin(\lambda x)][/tex]

But the book claimed the solution of y:[tex]y=C_{1}J_{\frac{1}{2}}[\sqrt{\frac{2}{\pi\lambda x}}sin(\lambda x)] + C_{2}Y_{-\frac{1}{2}}[\sqrt{\frac{2}{\pi\lambda x}}sin(\lambda x)][/tex]

Question is why do I have to use Y in the second solution instead of both J to solve the boundary value problem? I always learn that if you get the two independent solution, that would be the answers!
 
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  • #2
I forgot to mention, I understand [tex]J_{-p}[/tex] is independent only if p is not an integer, you have to use Bessel function of second kind...Y as the second linearly independent solution. In that case, you are stuck with Y.

But in here, you don't need to. I guess my ultimate question is why people even use Y when p is not an integer to make their life more complicated!
 

1. What is the parametric Bessel's equation?

The parametric Bessel's equation is a second-order differential equation that arises in mathematical physics and engineering. It is used to describe the behavior of certain physical systems, such as heat transfer and vibration, and is named after the mathematician Friedrich Bessel.

2. What is the solution to the parametric Bessel's equation?

The solution to the parametric Bessel's equation is a set of functions called Bessel functions, denoted by Jα(x) and Yα(x). These functions are essential in solving various problems in mathematical physics, especially those involving circular and cylindrical symmetry.

3. What are the applications of the solution to parametric Bessel's equation?

The solution to parametric Bessel's equation has various applications in physics, engineering, and mathematics. They are used to solve problems involving heat transfer, sound waves, electromagnetic fields, and fluid flow. They also play a crucial role in the theory of special functions and differential equations.

4. How do you find the solution to the parametric Bessel's equation?

The solution to parametric Bessel's equation can be found using various methods, such as power series, Frobenius method, and integral representations. The choice of method depends on the specific problem and its boundary conditions. Numerical methods, such as the shooting method, can also be used to approximate the solution.

5. Can the solution to the parametric Bessel's equation be expressed in closed form?

Yes, the solution to parametric Bessel's equation can be expressed in closed form for certain values of the parameters. For example, when the parameter α is an integer, the Bessel functions reduce to polynomial functions. However, for non-integer values of α, the Bessel functions cannot be expressed in closed form and must be evaluated numerically.

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