Let cos$$^{ - 1}\left( {{y \over b}} \right) = \log {\left( {{x \over n}} \right)^n}$$. Then

Let f be a differentiable function with $$\mathop {\lim }\limits_{x \to \infty } f(x) = 0.$$ If $$y' + yf'(x) - f(x)f'(x) = 0$$, $$\mathop {\lim }\limits_{x \to \infty } y(x) = 0$$, then (where $$y \equiv {{dy} \over {dx}})$$

If $$x\sin \left( {{y \over x}} \right)dy = \left[ {y\sin \left( {{y \over x}} \right) - x} \right]dx,\,x > 0$$ and $$y(1) = {\pi \over 2}$$, then the value of $$\cos \left( {{y \over x}} \right)$$ is

The differential equation of the family of curves y = e^x (A cos x + B sin x) where, A, B are arbitrary constants is