Question

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

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

Answer 1

Answer: (E)

We have, $ y\frac{dy}{dx}+x=c $ On it can be rewritten as $ ydy=(c-x)dx $ On integrating both sides, we get $ \int{y\,\,dy}=\int{(c-x)}\,dx $ $ \frac{{{y}^{2}}}{2}=cx-\frac{{{x}^{2}}}{2}+{{c}_{1}} $ $ \Rightarrow \,\,{{x}^{2}}+{{y}^{2}}\,-2cx={{c}_{1}} $ Which represents a family of circles whose centre is (c, 0) i. e., on the $ x- $ axis.