Differential equation whose general solution is $y=c_{1}x+\frac{c_2}{x}$ for all values of c₁ and c₂ is:

$y=c_{1}x+\frac{c_2}{x}$ ... (i)

There are two arbitrary constants. To eliminate these constants, we need to differentiate (i) twice. Differentiating (i) with respect to x,

$\frac{dy}{dx}=c_1-\frac{c_2}{x^2}$  ... (ii)

Again differentiating with respect to x,

$\frac{d^{2}y}{d{x}^2}=\frac{2c_2}{x^3}$  ... (iii)

From (iii), $c_2=\frac{x^3}{2}\frac{d^{2}y}{d{x}^2}$ and from (ii), $c_1=\frac{dy}{dx}+\frac{c_2}{x^2}$;

$\therefore c_1=\frac{dy}{dx}+\frac{x}{2}\frac{d^{2}y}{d{x}^2}$

From (i),  $y=\left(\frac{dy}{dx}+\frac{x}{2}\cdot \frac{d^{2}y}{d{x}^2}\right)x+\frac{x^2}{2}\cdot \frac{d^{2}y}{d{x}^2}$

$\begin{array}{l}\Rightarrow y=x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}\\ \therefore \frac{d^{2}y}{d{x}^2}+\frac{1}{x}\frac{dy}{dx}-\frac{y}{x^2}=0\end{array}$