If x = sin t, y = sin 2t, prove that $$(1 - {x^2}){{{d^2}y} \over {d{x^2}}} - x{{dy} \over {dx}} + 4y = 0$$

If f is differentiable at x = a, find the value of $$\mathop {\lim }\limits_{x \to a} {{{x^2}f(a) - {a^2}f(x)} \over {x - a}}$$

If $${{dy} \over {dx}} + \sqrt {{{1 - {y^2}} \over {1 - {x^2}}}} = 0$$, prove that $$x\sqrt {1 - {y^2}} + y\sqrt {1 - {x^2}} = A$$, where A is a constant.

Let R be the set of real numbers and f : R $$\to$$ R be such that for all x, y $$\in$$ R, $$|f(x) - f(y)| \le |x - y{|^3}$$. Prove that f is a constant function.