Question

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

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

Answer 1

The correct option is A a hyperbola

$\frac{dy}{dx}=\frac{x^2+y^2+1}{2xy}$

$2y\frac{dy}{dx}=x+\frac{y^2}{x}+\frac{1}{x}$

$2y\frac{dy}{dx}-\frac{y^2}{x}=x+\frac{1}{x}$

$y^2=u=2y\frac{dy}{dx}=\frac{du}{dx}$

$\frac{du}{dx}-\frac{u}{x}=x+\frac{1}{x}$

$\frac{du}{dx}+Pu=Q$

$\frac{1}{x}\frac{du}{dx}-\frac{u}{x^2}=1+\frac{1}{x^2}$

$\int d\left(\frac{u}{x}\right)=\int (1+\frac{1}{x^2})dx\Rightarrow \frac{u}{x}x-x^{-1}+C$

$\Rightarrow \frac{y^2}{x}=x-\frac{1}{x}+C$

$\Rightarrow y^2=x^2-1+x$