Question

Solution of the equation $x{\left(\frac{dy}{dx}\right)}^2+2\sqrt{xy}\frac{dy}{dx}+y=0$ is:

• A. $x+y=a$
• B. $\sqrt{x}-\sqrt{y}=\sqrt{a}$
• C. $x^2+y^2=a^2$
• D. $\sqrt{x}+\sqrt{y}=\sqrt{a}$
• E. $x^2-y^2=a^2$

Answer 1

Answer: (D)

$x{\left(\frac{dy}{dx}\right)}^2+2\sqrt{xy}\frac{dy}{dx}+y=0$ $\Rightarrow$ ${\left(\sqrt{x}\frac{dy}{dx}+\sqrt{y}\right)}^2=0$ On integrating both sides $\int \frac{1}{\sqrt{y}}dy+\int \frac{1}{\sqrt{x}}dx=0$ $\Rightarrow$ $2\sqrt{y}+2\sqrt{x}=c_1$ $\Rightarrow$ $\sqrt{x}+\sqrt{y}=\frac{c_1}{2}$ which is similar to $\sqrt{x}+\sqrt{y}=\sqrt{a}.$ $\therefore$ Solution of given differential equation is $\sqrt{x}+\sqrt{y}=\sqrt{a}$