Example 2. Equation

$$
\frac{d y}{d x}=\frac{m^{2}}{x^{4}}-y^{2}, \quad m=\text { const }
$$

has particular solutions

$$
y_{1}=\frac{1}{x}+\frac{m}{x^{2}}, \quad y_{2}=\frac{1}{x}-\frac{m}{x^{2}}
$$

Find its general integral.
Solution. Using formula (7), we obtain the general integral of the original equation

$$
\frac{y-y_{1}}{y-y_{2}}=C e^{-\int\left(2 m / x^{2}\right) d x}, \quad \text { from which } \quad \frac{x^{2} y-x-m}{x^{2} y-x+m}=C e^{2 m / x}
$$

## Problems for Independent Solution

Integrate the following Riccati equations, knowing their particular solutions:

232. $y^{\prime} e^{-x}+y^{2}-2 y e^{x}=1-e^{2 x}, y_{1}=e^{x}$.
233. $y^{\prime}+y^{2}-2 y \sin x+\sin ^{2} x-\cos x=0, y_{1}=\sin x$.
234. $x y^{\prime}-y^{2}+(2 x+1) y=x^{2}+2 x, y_{1}=x$.
235. $x^{2} y^{\prime}=x^{2} y^{2}+x y+1, y_{1}=-\frac{1}{x}$.
236. Find the general integral of the Riccati equation when the ratio of coefficients does not depend on $x$, i.e., $a(x): b(x): c(x)=m: n: p(m, n, p$ - constants). 237. Prove that the Riccati equation retains its form under any transformation of the independent variable $x=\varphi(t)$, where $\varphi(t)$ is any continuously differentiable function defined in the interval $\left(t_{0}, t_{1}\right)$, and $\varphi^{\prime}(t) \neq 0$ on $\left(t_{0}, t_{1}\right)$.

## § 10. Formulation of Differential Equations of Families of Curves. Problems on Trajectories

$1^{\circ}$. Formulation of differential equations of families of curves. Let the equation of a one-parameter family of plane curves be given by

$$
y=\varphi(x, a) \quad(a-\text { parameter })
$$

Differentiating (I) with respect to $x$, we find

$$
y^{\prime}=\varphi_{x}^{\prime}(x, a)
$$

Eliminating the parameter $a$ from (1) and (2), we obtain the differential equation

$$
F\left(x, y, y^{\prime}\right)=0
$$

expressing a property common to all curves of the family (1). Equation (3) will be the desired differential equation of the family (1).

If a one-parameter family of curves is defined by the equation

$$
\Phi(x, y, a)=0
$$

then the differential equation of this family is obtained by eliminating the parameter $a$ from the equations

$$
\left\{\begin{array}{l}
\Phi(x, y, a)=0 \\
\frac{\partial \Phi}{\partial x}+\frac{\partial \Phi}{\partial y} y^{\prime}=0
\end{array}\right.
$$

Now suppose we have the relation

$$
\Phi\left(x, y, a_{1}, a_{2}, \ldots, a_{n}\right)=0
$$

where $a_{1}, a_{2}, \ldots, a_{n}$ are parameters. Differentiating (4) $n$ times with respect to $x$ and eliminating the parameters $a_{1}, a_{2}, \ldots, a_{n}$ from (4) and the obtained equations, we arrive at a relation of the form

$$
F\left(x, y, y^{\prime}, y^{\prime \prime}, \ldots, y^{(n)}\right)=0
$$

This differential equation represents the given $n$-parameter family of curves (4) in the sense that (4) is the general integral of equation (5).