Question

If y=√x+1/√x show that 2x d y /dx +y =2√x

Answer 1

$\textbf{Given:}$

$y=\sqrt{x}+\frac{1}{\sqrt{x}}$

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

$\text{Differentiate with respect to x}$

$\displaystyle\frac{dy}{dx}=\frac{1}{2}x^{\frac{-1}{2}}-\frac{1}{2}x^{\frac{-3}{2}}$

$\displaystyle\frac{dy}{dx}=\frac{1}{2\sqrt{x}}-\frac{1}{2\,x\sqrt{x}}$

$\text{Now,}$

$\displaystyle\,2x\,\frac{dy}{dx}+y$

$=\displaystyle,2x(\frac{1}{2\sqrt{x}}-\frac{1}{2\,x\sqrt{x}})+\sqrt{x}+\frac{1}{\sqrt{x}}$

$=\displaystyle\frac{2x}{2\sqrt{x}}-\frac{2x}{2\,x\sqrt{x}}+\sqrt{x}+\frac{1}{\sqrt{x}}$

$=\displaystyle\sqrt{x}-\frac{1}{\sqrt{x}}+\sqrt{x}+\frac{1}{\sqrt{x}}$

$=\displaystyle\,2\,\sqrt{x}$

$\implies\boxed{\bf\,2x\,\frac{dy}{dx}+y=2\,\sqrt{x}}$