Question

lf $f\left(x\right)={\sin }^{-1}x$ then $(1+x^2)f^{(n+2)}\left(x\right)+2x(n+1)f^{(n+1)}\left(x\right)+n(n+1)f^{\left(n\right)}\left(x\right)\forall (n\ge 1)$ is equal to

• A. $0$
• B. $f\left(x\right)$
• C. $-f\left(x\right)$
• D. $2f\left(x\right)$

Answer 1

The correct option is A $0$

Given, $f\left(x\right)={\sin }^{-1}x$

$f^{\prime }\left(x\right)=\frac{1}{1+x^2}$

On Cross-multiplication, we get

$(1+x^2)f^{\prime }\left(x\right)=1$

$\Rightarrow 2x{f}^{\prime }\left(x\right)+(1+x^2)f^{\prime \prime }\left(x\right)=0$

$\Rightarrow 2f^{\prime }\left(x\right)+2x{f}^{\prime \prime }\left(x\right)+2x{f}^{\prime \prime }\left(x\right)+(1+x^2)f{{}^{\prime \prime }}^{\prime }\left(x\right)=0$

$\Rightarrow (1+x^2)f^4\left(x\right)+2x{f}^3\left(x\right)+4x{f}^3\left(x\right)+6f^{\prime \prime }\left(x\right)=0$

$\Rightarrow (1+x^2)f^{n+2}\left(x\right)+2x(n+1)f^{n+1}\left(x\right)+n(n+1)f^n\left(x\right)=0$