Question

The curve amongst the family of curves, represented by the differential equation, $(x^2 - y^2)dx + 2xy dy = 0 $ which passes through (1,1) is :

• A. A circle with centre on the $y$-axis
• B. A circle with centre on the $x$-axis
• C. An ellipse with major axis along the $y$-axis
• D. A hyperbola with transverse axis along the $x$-axis

Answer 1

Answer: (B)

$\left(x^{2}-y^{2}\right)dx+2xy dy =0 $
$ \frac{dy}{dx} = \frac{y^{2}-x^{2}}{2xy} $
Put $y=vx \Rightarrow \frac{dy}{dx} =v +x \frac{dv}{dx} $
Solving we get,
$ \int \frac{2v}{v^{2}+1} dv =\int -\frac{dx}{x} $
$ln\left(v^{2}+1\right)=-ln x+C $
$\left(y^{2}+x^{2} \right)=Cx $
$ 1+ 1 = C \Rightarrow C = 2$
$ y^{2}+x^{2} =2x $