Question

lf $y=e^{a{\cos }^{-1}x}$ then $(1-x^2)y^{\prime \prime }-x{y}^{\prime }=$

• A. $ay$
• B. $-a^{2}y$
• C. $-ay$
• D. $a^{2}y$

Answer 1

The correct option is D $a^{2}y$

$y^1=e^{a{\cos }^{-1}x}\left(\frac{-a}{\sqrt{1+x^2}}\right)$

$y^{11}=e^{a{\cos }^{-1}x}\left(\frac{a}{\sqrt{1+x^2}}\right)+e^{a{\cos }^{-1}x}\left(\frac{-ax}{\sqrt{(1-x^2{)}^{\frac{3}{2}}}}\right)$

$(1-x^2)y^{\prime \prime }=a^{2}y+y^{\prime }x$

$(1-x^2)y^{\prime \prime }-y^{\prime }x=a^{2}y$