The differential equation y frac{dy}{dx} + x | vedclass

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(B) Given differential equation is $y \frac{dy}{dx} + x = c$. Separating the variables,we get $y \, dy = (c - x) \, dx$. Integrating both sides,we hav

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a family of circles whose centres are on the $x$-axis

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(B) Given differential equation is $y \frac{dy}{dx} + x = c$. Separating the variables,we get $y \, dy = (c - x) \, dx$. Integrating both sides,we have $\int y \, dy = \int (c - x) \, dx$. This gives $\frac{y^2}{2} = cx - \frac{x^2}{2} + k$,where $k$ is the constant of integration. Multiplying by $2$,we get $y^2 = 2cx - x^2 + 2k$,which simplifies to $x^2 - 2cx + y^2 = 2k$. Completing the square for $x$,we get $(x^2 - 2cx + c^2) + y^2 = 2k + c^2$. Thus,$(x - c)^2 + y^2 = R^2$,where $R^2 = 2k + c^2$. This is the equation of a family of circles with centres at $(c, 0)$,which lie on the $x$-axis.

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

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