Question

The curve amongst the family of curves, represented by the differential equation, $(x^2-y^2)dx+2xydy=0$ 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)

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