Question

The differential equation $y\frac{dy}{dx}+x=c$ represents

• A. a family of hyperbolas
• B. a family of circles whose centers are on the y-axis
• C. a family of parabolas
• D. a family of circles whose centers are on the x-axis

Answer 1

The correct option is D

a family of circles whose centers are on the x-axis

Explanation for correct option

Finding what the differential equation represents

Given differential equation, $y\frac{dy}{dx}+x=c$

Simplifying,
$$\begin{gathered} y\left(\frac{dy}{dx}\right)+x=c \\ \Rightarrow y\left(\frac{dy}{dx}\right)=c-x \\ \Rightarrow ydy=(c-x)dx. \\ \Rightarrow ydy=cdx-xdx \end{gathered}$$

On integrating both sides,
$$\begin{gathered} \int \mathrm{ydy}=\int \mathrm{cdx}-\int \mathrm{xdx} \\ \Rightarrow \frac{{\mathrm{y}}^2}{2}=\mathrm{cx}-\frac{{\mathrm{x}}^2}{2}+\mathrm{k}\left[\mathrm{k}=constant\right] \\ \Rightarrow \frac{{\mathrm{x}}^2}{2}+\frac{{\mathrm{y}}^2}{2}=\mathrm{cx}+\mathrm{k} \\ \Rightarrow {\mathrm{x}}^2+{\mathrm{y}}^2=2\mathrm{cx}+\mathrm{k} \\ \Rightarrow {\mathrm{x}}^2+{\mathrm{y}}^2-2\mathrm{cx}-\mathrm{k}=0 \end{gathered}$$

The equation represents a family of circles whose centre is on the x-axis as there is no $\mathrm{y}$ variable.

Hence, the correct answer is Option (D).