Question

If $y = {\left( {{{\sin }^{ - 1}}x} \right)^2}$ then show that$\left( {1 - {x^2}} \right)\dfrac{{{d^2}y}}{{d{x^2}}} - x\dfrac{{dy}}{{dx}} = 2$

Answer 1

Hint -Differentiate $y = {\left( {{{\sin }^{ - 1}}x} \right)^2}$ with respect to x and adjust the obtained result. When you again differentiate the adjusted result, you’ll get the answer.

Complete step-by-step answer:
Given,$y = {\left( {{{\sin }^{ - 1}}x} \right)^2}$
On differentiating y with respect to x, we get$\dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}{\left( {{{\sin }^{ - 1}}x} \right)^2} \Rightarrow 2{\sin ^{ - 1}}x \times \dfrac{1}{{\sqrt {1 - {x^2}} }}$ --- (i)
$ \Rightarrow \left( {\sqrt {1 - {x^2}} } \right)\dfrac{{dy}}{{dx}} = 2{\sin ^{ - 1}}x$ --- (ii)
On differentiating eq. (ii) w.r.t. x, we get
$
  \left( {\sqrt {1 - {x^2}} } \right)\dfrac{{{d^2}y}}{{d{x^2}}} - \dfrac{{2x}}{{2\sqrt {1 - {x^2}} }}\dfrac{{dy}}{{dx}} = \dfrac{2}{{\sqrt {1 - {x^2}} }} \\
   \Rightarrow \left( {1 - {x^2}} \right)\dfrac{{{d^2}y}}{{d{x^2}}} - x\dfrac{{dy}}{{dx}} = 2 \\
 $
Hence Proved.
Note: You can also solve this by directly differentiating eq. (i) and then solving and adjusting the equation. Then put the value of eq. (i) in the last step and you will obtain the answer.