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} $