The differential equation of curve given by $\mathrm{y}=\mathrm{a} \cos \left(\frac{\mathrm{n}}{\mathrm{x}}+\mathrm{b}\right)$ (a,b are arbitrary constants)
 

$\mathrm{y}=\mathrm{a} \cos \left(\frac{\mathrm{n}}{\mathrm{x}}+\mathrm{b}\right)$

diff w.r.t  x

$$\begin{gathered} \frac{\mathrm{dy}}{\mathrm{dx}}=-\mathrm{a} \sin \left(\frac{\mathrm{n}}{\mathrm{x}}+\mathrm{b}\right)·\left(\frac{-\mathrm{n}}{{\mathrm{x}}^2}\right) \\ {\mathrm{x}}^2 \frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{an} \sin \left(\frac{\mathrm{n}}{\mathrm{x}}+\mathrm{b}\right) -\left(1\right) \end{gathered}$$

Again diff w.r.t

$$\begin{gathered} {\mathrm{x}}^2 \frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}+\frac{\mathrm{dy}}{\mathrm{dx}} \left(2\mathrm{x}\right)=\mathrm{an} \cos \left(\frac{\mathrm{n}}{\mathrm{x}}+\mathrm{b}\right) \left(\frac{-\mathrm{n}}{{\mathrm{x}}^2}\right) \\ \Rightarrow {\mathrm{x}}^4 \frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}+2{\mathrm{x}}^3 \frac{\mathrm{dy}}{\mathrm{dx}}=-{\mathrm{an}}^2 \cos \left(\frac{\mathrm{n}}{\mathrm{x}}+\mathrm{b}\right) \\ \Rightarrow {\mathrm{x}}^4{\mathrm{y}}_2+2{\mathrm{x}}^3{\mathrm{y}}_1=-{\mathrm{n}}^2\mathrm{y} \\ \Rightarrow \boxed{ {\mathrm{x}}^4{\mathrm{y}}_2+2{\mathrm{x}}^3{\mathrm{y}}_1+{\mathrm{n}}^2\mathrm{y}=0 } \end{gathered}$$