Differential Equations
Formation of Differential Equations
Premium Question
Grade 12
Question:
The differential equation of the family of circles passing through the origin and having their centres on the $x$-axis is:
(1) $\frac{dy}{dx} = \frac{x^2 - y^2}{2xy}$
(2) $\frac{dy}{dx} = \frac{y^2 - x^2}{2xy}$
(3) $\frac{dy}{dx} = \frac{2xy}{x^2 - y^2}$
(4) $\frac{dy}{dx} = \frac{x^2 + y^2}{2xy}$
Step-by-Step Solution
Key Concept: Write the family as $x^2 + y^2 = 2cx$, differentiate once to express $c$ in terms of $x, y, \frac{dy}{dx}$, and substitute this back to eliminate the arbitrary constant $c$.
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Correct Answer: (2)