We consider the curve

$\begin{array}{}\text{(1)}& \alpha (x)=(x,y(x)),\end{array}$

with tangent vector

$\begin{array}{}\text{(2)}& {\alpha}^{\prime}(x)=(1,{y}^{\prime}(x))=\mathbf{i}+{y}^{\prime}(x)\mathbf{j};\end{array}$

if

$\begin{array}{}\text{(3)}& F(x,y)=P(x,y)\mathbf{i}+Q(x,y)\mathbf{j}\end{array}$

is tangent to $\alpha (x)$ at (x,y), then ${\alpha}^{\prime}(x)$ is collinear with F(x,y); that is, there is some

$\begin{array}{}\text{(4)}& 0\ne \beta \in \mathbb{R}\end{array}$

with

$\begin{array}{}\text{(5)}& {\alpha}^{\prime}(x)=\beta F(x,y);\end{array}$

that is, by virtue of (2) and (3),

$\begin{array}{}\text{(6)}& \mathbf{i}+{y}^{\prime}(x)\mathbf{j}=\beta P(x,y)\mathbf{i}+\beta Q(x,y)\mathbf{j};\end{array}$

comparing coefficients yields

$\begin{array}{}\text{(7)}& \beta P(x,y)=1,\end{array}$

and

$\begin{array}{}\text{(8)}& \beta Q(x,y)={y}^{\prime}(x);\end{array}$

we observe that (7) implies $P(x,y)\ne 0$, hence we have

$\begin{array}{}\text{(9)}& \beta ={\displaystyle \frac{1}{P(x,y)}},\end{array}$

and combining this with (8) we find

$\begin{array}{}\text{(10)}& {y}^{\prime}(x)=\beta Q(x,y)={\displaystyle \frac{1}{P(x,y)}}Q(x,y)={\displaystyle \frac{Q(x,y)}{P(x,y)}}.\end{array}$

Now with

$\begin{array}{}\text{(11)}& F(x,y)=y\mathbf{i}+x\mathbf{j},\end{array}$

we obtain

$\begin{array}{}\text{(12)}& {y}^{\prime}(x)={\displaystyle \frac{x}{y}},\end{array}$

or

$\begin{array}{}\text{(13)}& y{y}^{\prime}(x)=x;\end{array}$

we observe that

$\begin{array}{}\text{(14)}& {\displaystyle \frac{1}{2}}({y}^{2}(x){)}^{\prime}=y{y}^{\prime}(x);\end{array}$

(13) may thus be written as

$\begin{array}{}\text{(15)}& {\displaystyle \frac{1}{2}}({y}^{2}(x){)}^{\prime}={\displaystyle \frac{1}{2}}({x}^{2}{)}^{\prime},\end{array}$

or

$\begin{array}{}\text{(16)}& {\displaystyle \frac{1}{2}}({y}^{2}(x)-{x}^{2}{)}^{\prime}=0,\end{array}$

whence

$\begin{array}{}\text{(17)}& ({y}^{2}(x)-{x}^{2}{)}^{\prime}=0,\end{array}$

which implies that

$\begin{array}{}\text{(18)}& {y}^{2}(x)-{x}^{2}=C,\phantom{\rule{thickmathspace}{0ex}}\text{a constant};\end{array}$ a constant;(18)

the field lines of (11) are thus the curves

$\begin{array}{}\text{(19)}& {y}^{2}-{x}^{2}=C,\end{array}$

which is a family of hyperbolas in ${\mathbb{R}}^{2}$

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