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**Lancet** I'm working with the Cauchy-Euler equation... In the direction of

$\displaystyle a \frac{d^2y}{dt^2} + b \frac{dy}{dt} + cy = 0$

...being transformed into

$\displaystyle Ax^2 \frac{d^2y}{dx^2} + Bx \frac{dy}{dx} + Cy = 0$

by setting

$\displaystyle t = ln(x)$

... I have been able to prove how

$\displaystyle \frac{dy}{dt} = x \frac{dy}{dx}$

and

$\displaystyle \frac{d^2y}{dt^2} = x^2 \frac{d^2y}{dx^2} + x \frac{dy}{dx}$

Oddly enough, what I'm getting stuck on, is how $\displaystyle a, b, c$ relate to $\displaystyle A, B, C$.

It seems to me (please let me know if I'm wrong) that

$\displaystyle b = B$

$\displaystyle c = C$

...but I'm not exactly sure what's going on with $\displaystyle a$.

Can someone help me figure out what I'm missing?