- #1

s3a

- 818

- 8

## Homework Statement

How do I get d^2 y/dx^2 for a Cauchy-Euler, differential equation?

Basically, how do I derive d^2 y/dx^2, as given in the following link (since I don't want to just memorize that equation)?:

http://www.sosmath.com/diffeq/second/euler/euler.html

## Homework Equations

* Cauchy-Euler, differential Equation.

* x = e^t

* Differentiating.

* Chain rule

## The Attempt at a Solution

I do get how to derive dy/dx.

When I try to derive d^2 y/dx^2, I get a very close result, but it's different.

Could someone please tell me where my mistake is, or if I'm doing something completely wrong, tell me what it is I am doing wrong?

__Here is my work.:__

x = e^t

dy/dx = 1/x dy/dt

d/dx (dy/dx) = d/dx(1/x dy/dt)

d^2 x/dx^2 = d/dx (1/x) dy/dt + (1/x) d/dx (dy/dt)

d^2 x/dx^2 = d/dx (1/x) dy/dt + (1/x) d/dt (dy/dx)

d^2 x/dx^2 = d/dx (1/x) dy/dt + (1/x) d/dt (1/x dy/dt)

d^2 x/dx^2 = d/dx (1/x) dy/dt + (1/x) d/dt (e^(-t) dy/dt)

d^2 x/dx^2 = (-1/x^2) dy/dt + [e^(-t)] d/dt (e^(-t) dy/dt)

d^2 x/dx^2 = [-e^(-2t)] dy/dt + [e^(-t)] (-[e^(-t)] [dy/dt] + [e^(-t)] [d^2 y/dt^2])

d^2 x/dx^2 = -e^(-2t) dy/dt – e^(-2t) dy/dt + e^(-2t) d^2 y/dt^2

d^2 x/dx^2 = -2e^(-2t) dy/dt + e^(-2t) d^2 y/dt^2

d^2 x/dx^2 = e^(-2t) [d^2 y/dt^2 - 2 dy/dt]

Any help in deriving d^2 y/dx^2 would be GREATLY appreciated!