Question

If y = (sin⁻¹ x)², prove that (1 − x²) $\frac{d^{2}y}{d{x}^2}-x\frac{dy}{dx}+p^{2}y=0$.

Answer 1

Here,

$$\begin{gathered} y={\left({\sin }^{-1}x\right)}^2 \\ \mathrm{Now}, \\ {y}_1=2{\sin }^{-1}x\frac{1}{\sqrt{1-x^2}} \\ \Rightarrow y_2=\frac{2}{1-x^2}+\frac{2x{\sin }^{-1}x}{{\left(1-x^2\right)}^{3/2}} \\ \Rightarrow y_2=\frac{2}{1-x^2}+\frac{2x{\sin }^{-1}x}{\left(1-x^2\right)\sqrt{1-x^2}} \\ \Rightarrow y_2=\frac{2}{1-x^2}+\frac{x{y}_1}{\left(1-x^2\right)} \\ \Rightarrow y_2\left(1-x^2\right)=2+x{y}_1 \\ \Rightarrow y_2\left(1-x^2\right)-x{y}_1-2=0 \end{gathered}$$

Hence proved.