Question

If $y=\sin \left(a{\sin }^{-1}x\right)$ then prove that $\left(1-x^2\right)y_2-x{y}_1+a^{2}y=0$

Answer 1

$y=\sin \left(a{\sin }^{-1}x\right)$

$y_1=\cos \left(a{\sin }^{-1}x\right)\frac{d}{dx}\left(a{\sin }^{-1}x\right)$

$y_1=a\cos \left(a{\sin }^{-1}x\right)\frac{d}{dx}\left({\sin }^{-1}x\right)$

$y_1=\frac{a\cos \left(a{\sin }^{-1}x\right)}{\sqrt{1-x^2}}$

$y_1=\frac{a\sqrt{1-y^2}}{\sqrt{1-x^2}}$

$y_1\sqrt{1-x^2}=a\sqrt{1-y^2}$

squaring both sides, we get

${y_1}^2(1-x^2)=a^2(1-y^2)$

Differentiating both sides, w.r.t $x$

$2y_1{y}_2(1-x^2)-2x{y_1}^2=-2a^{2}y{y}_1$

Dividing both sides by $2y_1$ we get

$(1-x^2)y_2-x{y}_1+a^{2}y=0$

Hence proved.