I was trying to solve a partial differential equation. I did all the basic steps easily to get this ordinary differential equation.
[imath]\displaystyle x\frac{d^2X}{dx^2} + \frac{dX}{dx} + \lambda X = 0[/imath]
This is not one of the equations that I am familiar with. Therefore, I had to cheat and I looked at the final solution and discovered that this equation can be transformed to the following Bessel equation.
[imath]\displaystyle r^2\frac{d^2X}{dr^2} + r\frac{dX}{dr} + \lambda r^2X = 0[/imath]
I have told topsquark before that the difficult part in solving a differential equation is how to transform it to one that you know how to solve. Can you suggest a substitution for [imath]x[/imath]? Let us see if it works!
FYI: The solution to the first equation is [imath]X(x)[/imath] and to the second equation is [imath]X(r)[/imath].
[imath]\displaystyle x\frac{d^2X}{dx^2} + \frac{dX}{dx} + \lambda X = 0[/imath]
This is not one of the equations that I am familiar with. Therefore, I had to cheat and I looked at the final solution and discovered that this equation can be transformed to the following Bessel equation.
[imath]\displaystyle r^2\frac{d^2X}{dr^2} + r\frac{dX}{dr} + \lambda r^2X = 0[/imath]
I have told topsquark before that the difficult part in solving a differential equation is how to transform it to one that you know how to solve. Can you suggest a substitution for [imath]x[/imath]? Let us see if it works!
FYI: The solution to the first equation is [imath]X(x)[/imath] and to the second equation is [imath]X(r)[/imath].