Question

If y = (sin⁻¹ x)², then (1 − x²)y₂ is equal to

(a) xy₁ + 2
(b) xy₁ − 2
(c) −xy₁+2
(d) none of these

Answer 1

(a) xy₁ + 2

Here,

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