Question

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?:
http://www.sosmath.com/diffeq/second/eul...

I do get how to derive dy/dx.

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

Answer 1

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

Answer 2

\[\begin{align*}\frac{dy}{dt}&=\frac{dy}{dx}\frac{dx}{dt}\\
&=e^t\frac{dy}{dx}\\
e^{-t}\frac{dy}{dt}&=\frac{dy}{dx}
\end{align*}\]
\[\begin{align*}\frac{d^2y}{dt^2}&=\frac{d}{dt}\left[\frac{dy}{dt}\right]\\
&=\frac{d}{dt}\left[e^{t}\frac{dy}{dx}\right]\\
&=e^t\frac{d}{dt}\left[\frac{dy}{dx}\right]+\frac{d}{dt}\left[e^t\right]\frac{dy}{dx}\\
&=e^t\frac{d}{dt}\left[y'(x(t))\right]+e^t\frac{dy}{dx}\\
&=e^t\left[y''(x(t))\cdot x'(t)\right]+e^t\frac{dy}{dx}\\
&=e^t\left[\frac{d^2y}{dx^2}\frac{dx}{dt}\right]+\frac{dy}{dt}\\
&=e^te^t\frac{d^2y}{dx^2}+\frac{dy}{dt}\\
\frac{d^2y}{dt^2}&=e^{2t}\frac{d^2y}{dx^2}+\frac{dy}{dt}\\
e^{-2t}\left(\frac{d^2y}{dt^2}-\frac{dy}{dt}\right)&=\frac{d^2y}{dx^2}
\end{align*}\]

Answer 3

Thank you very much!