y=px²+q $x^{- \frac{1}{2}}$ হলে 2x²y" -xy' কত হবে?

$\mathrm{y}=\mathrm{px}^{2}+\mathrm{qx}^{-\frac{1}{2}} ; \mathrm{y}^{\prime}=2 \mathrm{px}-\frac{1}{2} \mathrm{qx}^{-\frac{3}{2}} ; \mathrm{y}^{\prime \prime}=2 \mathrm{p}+\frac{3}{4} \mathrm{qx} \mathrm{x}^{-\frac{5}{2}}$

$\begin{array}{l} 2 x^{2} y^{\prime \prime}-x y^{\prime}=2 x^{2}\left(2 p+\frac{3}{4} q x^{-\frac{5}{2}}\right)-x\left(2 p x-\frac{1}{2} q x^{-\frac{3}{2}}\right) \\ =4 p x^{2}+\frac{3}{2} q x^{-\frac{1}{2}}-2 p x^{2}+\frac{1}{2} q x^{-\frac{1}{2}}=2 p x^{2}+2 q x^{-\frac{1}{2}}=2\left(p x^{2}+q x^{-\frac{1}{2}}\right)=2 y \end{array}$