Example 9. Form a differential equation whose general solution is

$$
y=C_{1} x+\frac{C_{2}}{x}
$$
Solution. Differentiating the given function, we get:

$$
y^{\prime}=C_{1}-\frac{C_{2}}{x^{2}}, y^{\prime \prime}=C_{1}-\frac{2 C_{2}}{x^{3}}
$$

From the equations

$$
y=C_{1} x+\frac{C_{2}}{x}, y^{\prime}=C_{1}-\frac{C_{2}}{x^{2}}
$$

we eliminate $C_{1}$. Multiplying the second equation by $x$ and subtracting the first from it, we get

$$
x y^{\prime}-y=-\frac{2 C_{2}}{x}
$$

Adding this equation to the equation $y^{\prime \prime}=\frac{2 C_{2}}{x^{3}}$, multiplied by $x^{2}$, we eliminate $C_{2}$:

$$
x y^{\prime}-y+y^{\prime \prime} x^{2}=0
$$

The obtained equation is the desired one.