Question

If y=eacos-1x,-1≤x≤1,show that (1-x2)$\frac{d^2y}{dx^2}$-x$\frac{dy}{dx}$-a2y=0

Answer 1

It is given that,y=eacos-1x
Taking logarithms on both sides, we obtain
logy=acos-1xloge
logy=acos-1x
Differentiating both sides with respect to x, we obtain
\frac{1}{y}$$\frac{dy}{dx}=a.$-\frac{1}{\sqrt{1-x^2}}$
$\Rightarrow$ $\frac{dy}{dx}$=-a$\frac{y^2}{{1-x^2}}$
By squaring both sides, we obtain
($\frac{dy}{dx}$)2=a2$\frac{y^2}{1-x^2}$
Again differentiating both sides with respect to x,we obtain
(1-x2)$\frac{d^2y}{dx^2}$-x$\frac{dy}{dx}$-a2y=0
Hence,proved