Example 2. Find the general solution of the equation $y^{\prime \prime \prime}=\frac{\ln x}{x^{2}}$ and identify the solution that satisfies the initial conditions $\left.y\right|_{x=1}=0,\left.y^{\prime}\right|_{x=1}=1$, $\left.y^{\prime \prime}\right|_{x=1}=2$
Solution. We integrate this equation sequentially three times:

$$
\begin{aligned}
y^{\prime \prime} & =\int \frac{\ln x}{x^{2}} d x=-\frac{\ln x}{x}-\frac{1}{x}+C_{1} \\
y^{\prime} & =-\frac{1}{2} \ln ^{2} x-\ln x+C_{1} x+C_{2} \\
y & =-\frac{x}{2} \ln ^{2} x+C_{1} \frac{x^{2}}{2}+C_{2} x+C_{3}
\end{aligned}
$$

We find the solution that satisfies the given initial conditions. Substituting the initial data $\left.y\right|_{x=1}=0,\left.y^{\prime}\right|_{x=1}=1,\left.y^{\prime \prime}\right|_{x=1}=2$ into (1), we get

$$
\frac{C_{1}}{2}+C_{2}+C_{3}=0, \quad C_{1}+C_{2}=1, \quad-1+C_{1}=2
$$

From this, $C_{1}=3, C_{2}=-2, C_{3}=\frac{1}{2}$. The desired solution is

$$
y=-\frac{x}{2} \ln ^{2} x+\frac{3}{2} x^{2}-2 x+\frac{1}{2}
$$