Example 3. Find the particular solution of the differential equation

$$
2 y\left(y^{\prime}+2\right)-x\left(y^{\prime}\right)^{2}=0
$$
Solution. A particular solution, if it exists, is determined by the system

$$
\left\{\begin{array}{r}
2 y\left(y^{\prime}+2\right)-x\left(y^{\prime}\right)^{2}=0 \\
2 y-2 x y^{\prime}=0
\end{array}\right.
$$

where the second equation (19) is obtained by differentiating (18) with respect to $y^{\prime}$. By eliminating $y^{\prime}$, we obtain the $p$-discriminant curve

$$
y^{2}+4 x y=0,
$$

which splits into two branches

$$
\begin{aligned}
& y=0 \\
& y=-4 x .
\end{aligned}
$$

By substitution, we verify that both functions are solutions to equation (18).

To determine whether the solutions (20) and (21) are singular or not, we find the envelope of the family

$$
C y-(C-x)^{2}=0,
$$

which is the general integral for (18).

We write the system for determining the $C$-discriminant curve

$$
\left\{\begin{aligned}
C y-(C-x)^{2} & =0 \\
y-2(C-x) & =0
\end{aligned}\right.
$$

from which, by eliminating $C$, we obtain

$$
y^{2}+4 x y=0, \text { or } y=0 \text { and } y=-4 x
$$

which coincides with (20) and (21). Since the conditions (11) and (12) are satisfied on the lines (20) and (21), we conclude that the lines $y=0$ and $y=-4 x$ are envelopes, and thus (20) and (21) are singular solutions of the given equation.

The integral curves (22) are parabolas $y=\frac{(C-x)^{2}}{C}$, and the lines $y=0$, $y=-4 x$ are the envelopes of this family of parabolas (Fig. 20).

![](https://cdn.mathpix.com/cropped/2024_05_22_a0612bb33256580e6dcdg-064.jpg?height=581&width=400&top_left_y=216&top_left_x=246)

Fig. 20

![](https://cdn.mathpix.com/cropped/2024_05_22_a0612bb33256580e6dcdg-064.jpg?height=643&width=395&top_left_y=152&top_left_x=747)

Fig. 21