Example 8. Find the differential equation for which the function $y=C_{1} x+C_{2}$, depending on two arbitrary constants, is the general solution.
Solution. Differentiating the given function twice, we eliminate the parameters $C_{1}$ and $C_{2}$:

$$
y^{\prime}=C_{1}, y^{\prime \prime}=0
$$

The equation $y^{\prime \prime}=0$ satisfies the condition of the problem.