Change of variable for leibniz notation

[progenitor]

Hi

I have a question for change of notation.

Quote from textbook:

As an example of a singular problem on a finite interval, consider the equation
xy'' + y' + λxy = 0, (6)
or
−(xy')' = λxy, (7)
on the interval 0 < x < 1, and suppose that λ > 0. This equation arises in the study of
free vibrations of a circular elastic membrane, and is discussed further in Section 11.5.
If we introduce the new independent variable t defined by t =sqrt(λ)x, then
dy/dx=sqrt(λ)*dy/dt,
d2y/dx2 = λ*d2y/dt2.
...

my question is the steps involved in deriving d2y/dx2 = λ d2y/dt2.

My understanding in deriving dy/dx=sqrt(λ)*dy/dt is as follows:

since dy/dt = dy/dx*dx/dt
and dx/dt ->(using differentials) dt = sqrt(λ)dx -> dx/dt = 1/sqrt(λ) -> dy/dx = sqrt(λ)*dy/dt.

However, how do you derive d2y/dx2 = λ d2y/dt2?
I get 1=1 which is no help at all?

Sorry if this is noobie, I'm kind of new.

Also, is there a online notepad for practicing mathjax synatax? I like pretty equations.

Thank you.

 

[tiny-tim]

welcome to pf!

hi progenitor! welcome to pf!

(have a square-root: √ and try using the X² icon just above the Reply box )

If we introduce the new independent variable t defined by t =sqrt(λ)x, then
…
… how do you derive d²y/dx² = λ d²y/dt²?

use the chain rule, d/dx = d/dt dt/dx = d/dt 1/√λ

(and you can practise your mathjax at https://www.physicsforums.com/mathjax/test/preview.html" )