Verify that${\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{cy}}^{\mathbf{2}}\mathbf{=}\mathbf{1}\mathbf{,}$ where c is an arbitrary non-zero constant, is a one-parameter family of implicit solutions to$\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{=}\frac{\mathbf{xy}}{{\mathbf{x}}^{\mathbf{2}}\mathbf{-}\mathbf{1}}$ and graph several of the solution curves using the same coordinate axes.

The required formula is$\frac{\mathbf{d}}{\mathbf{dx}}\mathbf{(}{\mathbf{x}}^{\mathbf{n}}\mathbf{)}\mathbf{=}\mathbf{n}{\mathbf{x}}^{\mathbf{n}\mathbf{-}\mathbf{1}}$.

$\mathrm{y}=\sqrt{\frac{1-{\mathrm{x}}^2}{\mathrm{c}}}$

$\begin{array}{l}\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1}{2\sqrt{\mathrm{c}}}{\left(1-{\mathrm{x}}^2\right)}^{\frac{-1}{2}}\times \left(-2\mathrm{x}\right)\\ \frac{\mathrm{dy}}{\mathrm{dx}}=-\frac{\mathrm{x}}{\sqrt{\mathrm{c}}}\times \frac{1}{\sqrt{1-{\mathrm{x}}^2}}\end{array}$

Multiplying and dividing the final differential equation obtained in Step 2 by $\sqrt{1-x^2}$:

role="math" localid="1663943533560" $\begin{array}{l}\frac{\mathrm{dy}}{\mathrm{dx}}=-\frac{\mathrm{x}}{\sqrt{\mathrm{c}}}\times \frac{1}{\sqrt{1-{\mathrm{x}}^2}}\frac{\sqrt{1-{\mathrm{x}}^2}}{\sqrt{1-{\mathrm{x}}^2}}\\ \frac{\mathrm{dy}}{\mathrm{dx}}=-\frac{\mathrm{x}}{\left(1-{\mathrm{x}}^2\right)}\left(\frac{\sqrt{1-{\mathrm{x}}^2}}{\sqrt{\mathrm{c}}}\right)\\ \frac{dy}{dx}=-\frac{xy}{\left(1-x^2\right)}\\ \frac{dy}{dx}=\frac{xy}{\left(x^2-1\right)}\end{array}$

Which is identical to the given differential equation.

Hence,$x^2+c{y}^2=1$ is a one-parameter family of implicit solutions to $\frac{dy}{dx}=\frac{xy}{x^2-1}$, for c as an arbitrary non-zero constant.

When$c=1$

$y=\sqrt{1-x^2}$ (Represented with red colour)

When $c=-1$

$y=\sqrt{-\left(1-x^2\right)}$ (Represented with a red-coloured dotted line)

When $c=2$

$y=\sqrt{\frac{1-x^2}{2}}$(Represented with blue colour)

When$c=-2$

$y=\sqrt{\frac{1-x^2}{\left(-2\right)}}$ (Represented with a blue-coloured dotted line)

When$c=3$

role="math" localid="1663944317705" $y=\sqrt{\frac{1-x^2}{3}}$(Represented with orange colour)

When$c=-3$

$y=\sqrt{\frac{1-x^2}{\left(-3\right)}}$ (Represented with orange-coloured dotted line)

Graph representing the solution curves corresponding to $c=\pm 1,\pm 2,\pm 3$.