Question

If y=e_acos-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-1xlog_e
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