Answers: 3
, then compute the derivatives of 
with respect to 
via the chain rule.
.![\dfrac{\mathrm d^2y}{\mathrm dx^2}=\dfrac{\mathrm d}{\mathrm dx}\left[\dfrac fx\right]=\dfrac{x\frac{\mathrm df}{\mathrm dx}-f}{x^2}](/tpl/images/0151/4356/50f86.png)
.![\dfrac{\mathrm d^3y}{\mathrm dx^3}=\dfrac{\mathrm d}{\mathrm dx}\left[\dfrac{g-f}{x^2}\right]=\dfrac{x^2\left(\frac{\mathrm dg}{\mathrm dx}-\frac{\mathrm df}{\mathrm dx}\right)-2x(g-f)}{x^4}](/tpl/images/0151/4356/790f4.png)
and its derivatives into the ODE gives a new one that is linear in 
and 
, so that the characteristic solution is
:
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