How to make a change of variable $y=x^2$ in the differential operator $frac{d^2}{d y^2}$?

Question

I have this differential operator
$$frac{d^2}{d y^2}qquad (1)$$
I want to make a change of variable $y=x^2$ in $(1)$ so that I get this differential operator
$$      frac{1}{4 x^2} frac{d^2 }{d x^2}-frac{1}{4 x^3}frac{d}{d x}  qquad (2)$$
How can I ask Mathematica to do this?

Alexei Boulbitch · Answer

Try this:
Clear[f];
D[f[y], {y, 2}] /. f -> (f[#^(1/2)] &) /. y -> x^2 // 
  Simplify[#, x > 0] & // Expand

(*  -(Derivative[1][f][x]/(4 x^3)) + (f^[Prime][Prime])[x]/(4 x^2)  *)

Have fun!

Michael E2 · Answer

Here's a counterexample to show that (1) applied to f[y] is not equivalent to (2) applied to f[x] under the substitution y == x^2, as it was stated in the OP in a previous edit and claimed in a comment:
Block[{f = Cos},
 {D[f[y], {y, 2}] /. y -> x^2,
   1/(4 x^2) D[f[x], {x, 2}] - 1/(4 x^3) D[f[x], x] // Simplify,
   1/(4 x^2) D[f[x^2], {x, 2}] - 1/(4 x^3) D[f[x^2], x] // Simplify
   } // Simplify]

How to make a change of variable $y=x^2$ in the differential operator $frac{d^2}{d y^2}$?

2 Answers

Add your own answers!

Ask a Question