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EEJ MAIN Mathematics QUESTION #6011
Question 1

Identify the curve from the family defined by $(x^2-y^2)\,dx + 2xy\,dy = 0$ that passes through the point $(1,1)$.

  • A circle centred on the $x$-axis
  • An ellipse with major axis along the $y$-axis
  • A circle centred on the $y$-axis✔️
  • A hyperbola with transverse axis along the $x$-axis
Correct Answer Explanation

Rewrite: $\frac{dy}{dx} = \frac{y^2-x^2}{2xy}$. This is a homogeneous ODE. Let $y=vx$:

$\frac{1-v^2}{2v}dx + x\,dv/... $

After solving, the general solution is $x^2 + y^2 = Cy$ (circles with centres on the $y$-axis).

Substituting $(1,1)$: $1+1=C\cdot1 \Rightarrow C=2$, giving $x^2+y^2=2y$, i.e. $x^2+(y-1)^2=1$.

This is a circle centred on the $y$-axis at $(0,1)$.