Question

If $y=\frac{1}{2}{\left({\sin }^{-1}x\right)}^2$, show that $\left(1-x^2\right)y_2-xy=1$.

Answer 1

$Y=\frac{1}{2}({\sin }^{-1}x{)}^2$

$y_1={\sin }^{-1}x\times \frac{1}{\sqrt{1}-x^2}$

$y_2=cfrac-11-x^2\times \frac{1}{2\sqrt{1-x^2}}\times (-2u){\sin }^{-1}+(\frac{1}{\sqrt{1-x^2}}{)}^2$

$(1-x^2)y_2=\frac{x{\sin }^{-1}x}{\sqrt{1-x^2}}+1$

$(1-x^2)y_2=x{y}_1+1$

$(1-x^2)y_2-x{y}_1=1$