Question

By introducing a new variable t, putting x=cos t , the expression is $(1-x^2)\frac{d^{2}y}{d{x}^2}-x\frac{dy}{dx}+y$transformed into :

• A.
• B.
• C.
• D.

Answer 1

The correct option is A

$t=co{s}^{-1}x\Rightarrow \frac{dt}{dx}=\frac{-1}{\sqrt{1-x^2}}$
$\frac{dy}{dx}=\frac{dy}{dx}.\frac{dt}{dx}$
$=-\frac{1}{\sqrt{1-x^2}}.\frac{dy}{dy}\Rightarrow \sqrt{1-x^2}\frac{dy}{dx}$
$=-\frac{dy}{dt}$
$\Rightarrow (1-x^2)=\frac{d^{2}y}{d{x}^2}-x\frac{dy}{dx}=\frac{d^{2}y}{d{t}^2}$