- #1

jonjacson

- 453

- 38

## Homework Statement

Transform the equation:

x

^{2}* d

^{2}y/dx

^{2}+ 2 * x * dy/dx + (a

^{2}/x

^{2})*y = 0

Using:

x=1/t

## Homework Equations

The differential of a function of several variables, and the common rules of differentiation.

https://en.wikipedia.org/wiki/Derivative

## The Attempt at a Solution

As I have x as function of t I can calculate dx as function of t and dt. So:

dx = (-1/t

^{2})* dt , equation 1

d

^{2}x = (2/t

^{3})*dt

^{2}, equation 2 (I considered d

^{2}t=0 because it is the independent variable)

To calculate dy/dx I symply change dx by its value at equation 1 , so I get:

dy/dx= dy/ (-1/t

^{2})*dt = -t

^{2}* (dy/dt)

(According to the book this is correct)

Now the problem is d

^{2}y/dx

^{2}

1.- First question, Why is it a mistake to simply substitute dx

^{2}for its value calculated in equation 2?

2.- According to the book I have to differenciate dy/dx which was equal to:

-t

^{2}dy/dt

I do it simply calculating the differential of a product:

(-2 * t * dt )* dy/dt -t

^{2}* d

^{2}y/dt

(I didn't differenciate dt, I assume d

^{2}t =0)

Simply dividing this value by dt I get what I expected to be the right result:

(-2*t) * dy/dt -t

^{2}* d

^{2}y/dt

^{2}

But according to the book, this is wrong, I am missing a term arising from dx/dt.

This is what the book does:

d

^{2}y/dx

^{2}= step 1 = d/dx (dy/dx) = step 2 = (d/dt ( dy/dx))/ ( dx/dt ) to get this result:- ( 2t dy/dt +t

^{2}* d

^{2}y/dt

^{2})(-t

^{2})

I understand the step 1 it simply says that the second derivative is the derivative of the first derivative, but the step 2 is a mistery to me and it is the step that creates the term I am missing that is the dx/dt in the denominator. I try to differenciate thinking in terms of differentials, so I can manipulate them in the expressions like they were algebraic quantities, maybe I don't understand the operator version of the derivative.

The last term -t

^{2}is what I didn't get in my calculation.

2.- Question two, in my previous calculation using differentials, Where I missed the dx/dt term?