Question

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

• A. a hyperbola
• B. a circle
• C. $y^2=x(1+x)-10$
• D. $(x-2{)}^2+(y-3{)}^2=5xy$

Answer 1

Answer: (A)

Given, $\frac{dy}{dx}=\frac{x^2+y^2+1}{2xy}$
$\Rightarrow 2xydy=\left(x^2+y^2+1\right)dx$
$\Rightarrow 2xydy-y^{2}dx=\left(x^2+1\right)dx$
$\Rightarrow xd\left(y^2\right)-y^{2}dx=\left(x^2+1\right)dx$
$\Rightarrow \frac{xd\left(y^2\right)-y^{2}dx}{x^2}=\left(1+\frac{1}{x^2}\right)dx$
$\Rightarrow d\left(\frac{y^2}{x}\right)=d\left(x-\frac{1}{x}\right)$
$\Rightarrow \frac{y^2}{x}=x-\frac{1}{x}+C$
$\Rightarrow y^2=x^2-1+Cx$
$\Rightarrow y^2={\left(x+\frac{C}{2}\right)}^2-1-\frac{C^2}{4}$
Clearly, it represents a hyperbola.