General equation for change of variable in a differential equation

[silmaril89]

I had a second order differential equation where $\psi$ is the unknown function and it is a function of $x$. We then introduced the following change of variable $x = \sqrt{\frac{\hbar}{m \omega}} \xi$. When all was said and done I found that,

$$\frac{d^2 \psi}{d \xi^2} = \bigg(\frac{dx}{d \xi}\bigg)^2 \frac{d^2\psi}{dx^2}$$

My question is, given an arbitrary change of variable for x and given an arbitrary order of the differential equation will the following formula always work?

$$\frac{d^n \psi}{d \xi^n} = \bigg(\frac{dx}{d\xi}\bigg)^n \frac{d^n \psi}{dx^n}$$

 

[lurflurf]

That will very not work. What you might have wanted is

$$\frac{d^n \psi}{d \xi^n} = \bigg(\frac{dx}{d\xi} \frac{d}{dx}\bigg)^n \psi$$

The trouble is that in general d/dx and dx/dxi do not commute and cannot be reordered. Try some examples like
x=exp(u)
x^2=u
and so on to see this.

 

[silmaril89]

Thanks for the response, but I don't seem to see the difference.

$$\frac{d^n \psi}{d \xi^n} = \bigg(\frac{dx}{d \xi} \frac{d}{dx} \bigg)^n \psi<br /> = \bigg(\bigg(\frac{dx}{d \xi} \bigg)^n \bigg(\frac{d}{dx} \bigg)^n \bigg) \psi<br /> = \bigg(\frac{dx}{d \xi} \bigg)^n \frac{d^n \psi}{dx^n}$$

Did I make any mistakes in the above? I'm not doing any reordering.

 

[lurflurf]

You are reordering
when you write

(a b)^3=a^3 b^3
a b a b a b=a a a b b b

you are making an implicit assumption that the order is not important
your change of variable equation holds when one variable is a constant multiple of the other

suppose
d/du=x d/dx
(d/du)^n=(x d/dx)^n
(d/du)^2=(x d/dx)^2=(x d/dx)(x d/dx)=x^2 (d/dx)^2+x d/dx
this is not equal to x^2 (d/dx)^2 because x d/dx is not 0

 

[silmaril89]

I see now. Thank you for clearing this up for me.