Question

If y = sin (m sin⁻¹ x), then (1 − x²) y₂ − xy₁ is equal to

Options

• m²y

• my

• −m²y

• none of these

Answer 1

(c)−m²y 

Here,

\[y = \sin\left( m \sin^{- 1} x \right)\]
\[ \Rightarrow y_1 = \cos\left( m \sin^{- 1} x \right)\frac{m}{\sqrt{1 - x^2}}\]
\[ \Rightarrow y_2 = - \sin\left( m \sin^{- 1} x \right)\frac{m^2}{\left( 1 - x^2 \right)} + \frac{mx\cos\left( m \sin^{- 1} x \right)}{\left( 1 - x^2 \right)^{3/2}}\]
\[ \Rightarrow y_2 = - \sin\left( m \sin^{- 1} x \right)\frac{m^2}{\left( 1 - x^2 \right)} + \frac{xm\cos\left( m \sin^{- 1} x \right)}{\left( 1 - x^2 \right) \times \sqrt{1 - x^2}}\]
\[ \Rightarrow y_2 = - \sin\left( m \sin^{- 1} x \right)\frac{m^2}{\left( 1 - x^2 \right)} + \frac{x y_1}{\left( 1 - x^2 \right)}\]
\[ \Rightarrow \left( 1 - x^2 \right) y_2 = - y m^2 + x y_1 \]
\[ \Rightarrow \left( 1 - x^2 \right) y_2 - x y_1 = - m^2 y\]