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Question

(A) Step $1$: Rewrite the differential equation as $(c + dy) dy = (a + bx) dx$.Step $2$: Integrate both sides: $\int (c + dy) dy = \int (a + bx) dx$.S

Vedclass · Accepted Answer

$a = c = 0, b + d = 0$

Vedclass · Answer

(A) Step $1$: Rewrite the differential equation as $(c + dy) dy = (a + bx) dx$.Step $2$: Integrate both sides: $\int (c + dy) dy = \int (a + bx) dx$.Step $3$: This gives $cy + \frac{d}{2} y^2 = ax + \frac{b}{2} x^2 + C$.Step $4$: Rearrange the terms: $\frac{b}{2} x^2 - \frac{d}{2} y^2 - ax + cy + C = 0$.Step $5$: For this to represent a circle centered at the origin, the coefficients of $x^2$ and $y^2$ must be equal, and the coefficients of $x$ and $y$ must be zero.Step $6$: Thus, $a = 0$, $c = 0$, and $\frac{b}{2} = -\frac{d}{2}$, which implies $b = -d$ or $b + d = 0$.

The solution of the differential equation $\frac{dy}{dx} = \frac{a + bx}{c + dy}$ represents a family of circles centered at the origin if...

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