Question

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

(a) m²y
(b) my
(c) −m²y
(d) none of these

Answer 1

(c)−m²y

Here,
$$\begin{gathered} y=\sin \left(m{\sin }^{-1}x\right) \\ \Rightarrow y_1=\cos \left(m{\sin }^{-1}x\right)\frac{m}{\sqrt{1-x^2}} \\ \Rightarrow y_2=-\sin \left(m{\sin }^{-1}x\right)\frac{m^2}{\left(1-x^2\right)}+\frac{mx\cos \left(m{\sin }^{-1}x\right)}{{\left(1-x^2\right)}^{3/2}} \\ \Rightarrow y_2=-\sin \left(m{\sin }^{-1}x\right)\frac{m^2}{\left(1-x^2\right)}+\frac{xm\cos \left(m{\sin }^{-1}x\right)}{\left(1-x^2\right)\times \sqrt{1-x^2}} \\ \Rightarrow y_2=-\sin \left(m{\sin }^{-1}x\right)\frac{m^2}{\left(1-x^2\right)}+\frac{x{y}_1}{\left(1-x^2\right)} \\ \Rightarrow \left(1-x^2\right)y_2=-y{m}^2+x{y}_1 \\ \Rightarrow \left(1-x^2\right)y_2-x{y}_1=-m^{2}y \end{gathered}$$