Question

The differential equation $\frac{dy}{dx}=\frac{\mathrm{x}\left(1+{\mathrm{y}}^2\right)}{\mathrm{y}\left(1+{\mathrm{x}}^2\right)}$ represents a family of

• A. Parabola
• B. Hyperbola
• C. Circle
• D. Ellipse

Answer 1

The correct option is B

Hyperbola

Explanation for correct option

Finding what a given differential equation represents

Given differential equation,

$$\begin{gathered} \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{x}\left(1+{\mathrm{y}}^2\right)}{\mathrm{y}\left(1+{\mathrm{x}}^2\right)} \\ \Rightarrow 2\left(\frac{\mathrm{y}}{1+{\mathrm{y}}^2}\right)\mathrm{dy}=2\left(\frac{\mathrm{x}}{1+{\mathrm{x}}^2}\right)\mathrm{dx} \end{gathered}$$

On Integrating we get,

$$\begin{gathered} \int \frac{2\mathrm{y}}{1+{\mathrm{y}}^2}\mathrm{dy}=\int \frac{2\mathrm{x}}{1+{\mathrm{x}}^2}\mathrm{dx} \\ \Rightarrow \int \frac{2\mathrm{x}}{1+{\mathrm{x}}^2}\mathrm{dx}=\log \left(1+{\mathrm{x}}^2\right) \\ \Rightarrow \log \left(1+{\mathrm{y}}^2\right)=\log \left(1+{\mathrm{x}}^2\right)+\log \mathrm{K}\left[\mathrm{K}=\mathrm{constant}\right] \\ \Rightarrow \left(1+{\mathrm{y}}^2\right)=\left(1+{\mathrm{x}}^2\right)\mathrm{K} \\ \Rightarrow \frac{{\mathrm{x}}^2}{1}-\frac{{\mathrm{y}}^2}{\mathrm{K}}=1-\mathrm{K} \end{gathered}$$

The equation $\frac{{\mathrm{x}}^2}{1}-\frac{{\mathrm{y}}^2}{\mathrm{K}}=1-\mathrm{K}$ is in the form of a hyperbola. Hence, the given differential equation represents a family of hyperbola.

Therefore, the correct answer is Option (B).