Question

The solution of $\frac{dy}{dx}=\frac{x^2+y^2+1}{2xy}$ satisfying y(1) = 0 is

• A. an ellipse with e $=\frac{1}{\sqrt{2}}$
• B. a hyperbola with e = $\sqrt{2}$
• C. a circle whose radius is 1
• D. a curve symmetric about origin

Answer 1

Answer: (B), (D)

The correct options are
B a hyperbola with e = $\sqrt{2}$
D a curve symmetric about origin

G.E. is $2y\frac{dy}{dx}-\frac{y^2}{x}=\frac{1}{x}+x$
Put $y^2=u\Rightarrow 2y\frac{dy}{dx}=\frac{du}{dx}$
$\therefore \frac{du}{dx}-\frac{1}{x}.u=\frac{1}{x}+x$
I.F. $=e^{-\int \frac{1}{x}dx}=\frac{1}{x}$
$\therefore$ sol. is $y^2=(x^2-1)+cx$
But y (1) = 0 $\Rightarrow$ c=0
$\Rightarrow x^2-y^2=1$ which is a rectangular hyperbola