Question

Solve the given differential equation.

$\frac{dy}{dx}+\frac{x\sqrt{(x^2+y^2)}-y^2}{xy}=0$

• A. $\sqrt{(x^2+y^2)}=xlogk/x$
• B. $\sqrt{(x^2+y^2)}=logk/x$
• C. $\sqrt{(x^2+y^2)}=x^{2}logk/x$
• D. none of these

Answer 1

The correct option is A $\sqrt{(x^2+y^2)}=xlogk/x$

Given, $\frac{dy}{dx}+\frac{x\sqrt{(x^2+y^2)}-y^2}{xy}=0$.

Put $y=vx\Rightarrow \frac{dy}{dx}=v+x\frac{dv}{dx}$

$\Rightarrow x\frac{dv}{dx}+\frac{\sqrt{(v^2+1)}}{v}=0$

$\Rightarrow \frac{1}{2}\frac{2v}{\sqrt{(1+v^2)}}dv+\frac{dx}{x}=0$

$\Rightarrow \frac{1}{2}\cdot 2\sqrt{(1+v^2)}+\log x=\log k$

$\Rightarrow \sqrt{(x^2+y^2)}=x\log k/x$