Change of variables in Seocnd order ODES

In summary, the modified bessel's equation, (x^2)y'' + (x)y' - (x^2 + p^2)y = 0, can be transformed into Bessel's equation by using the substitution t = ix. To do this, we start with dt = i dx and -t^2 = x^2, and then use the chain rule to
  • #1
phil ess
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I am looking through my course notes for mathematical physics, in preparation for the exam, and I've run into a concept that I can't figure out. It comes up first when talking about the modified bessel's equation

(x^2)y''+(x)y'-(x^2+p^2)y=0

And supposedly this can be transformed into bessel's euqation by using the substitution t=ix, but I can't figure out how this works. This is just the specific example that this issue came up in; I don't understand generally how to perform a change of variables for second order equations.

I start with dt=idx and -t^2=x^2, but how do I deal with y'' and y'?

If anyone can explain how change of variables works to me it would be greatly appreciated!
 
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  • #2
dt = i dx
dx = -i dt

dy/dx = dy/(-i dt) = i dy/dt

d²y/dx² = d(dy/dx)/dx = d(i dy/dt) / (-i dt) = - d²y/dt²
 

1. What is a change of variables in second order ODES?

A change of variables in second order ODES is a mathematical technique used to simplify and solve differential equations. It involves substituting a new variable in place of the independent or dependent variable in the differential equation, which can make the equation easier to solve or provide new insights into the problem.

2. Why do we use change of variables in second order ODES?

Change of variables in second order ODES allows us to transform a complex differential equation into a simpler form, making it easier to solve. It can also help us to identify new relationships and patterns within the problem, leading to a better understanding of the system being studied.

3. How do we choose the appropriate change of variables for a second order ODE?

The choice of change of variables depends on the specific problem being studied. Generally, we aim to transform the equation into a form that is easier to solve or analyze. This can include using substitutions, transformations, or even changing the independent or dependent variable. The best approach will depend on the structure and complexity of the initial equation.

4. Can change of variables be used for any type of second order ODE?

Yes, change of variables can be used for any type of second order ODE, including linear, non-linear, and non-homogeneous equations. However, the success of the technique will depend on the complexity of the initial equation and the chosen transformation.

5. Are there any limitations to using change of variables in second order ODEs?

While change of variables can be a powerful tool for solving differential equations, it may not always be applicable or effective in every situation. In some cases, the transformed equation may be more complex than the original, making it more difficult to solve. Additionally, some problems may require a more specialized approach or technique.

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