Euler Equations

Apr 2009
Q: An equation of the form

\(\displaystyle t^{2}\frac{d^{2}y}{dt^{2}}+\alpha\\t\frac{dy}{dt}+\beta\\y=0\), \(\displaystyle t>0\) with \(\displaystyle \alpha\) and \(\displaystyle \beta\) real constants, is called an Euler equation.

a) Let \(\displaystyle x=ln(t)\) and calculate \(\displaystyle \frac{d^{2}y}{dt^{2}}\) and \(\displaystyle \frac{dy}{dt}\) in terms of \(\displaystyle \frac{d^{2}y}{dx^{2}}\) and \(\displaystyle \frac{dy}{dx}\).

My question is, what funtion am I differentiating with respect x? Do I write \(\displaystyle y=t=e^{2}\) and differentiat that? What is the function y that I am differentiating?