Let $y(x)$ be the solution of the differential equation

$$ x^2 \frac{d y}{d x}+x y=x^2+y^2, \quad x>\frac{1}{e} $$

satisfying $y(1)=0$. Then the value of $2 \frac{(y(e))^2}{y\left(e^2\right)}$ is ____________.

For all x > 0, let y₁(x), y₂(x), and y₃(x) be the functions satisfying

$ \frac{dy_1}{dx} - (\sin x)^2 y_1 = 0, \quad y_1(1) = 5, $

$ \frac{dy_2}{dx} - (\cos x)^2 y_2 = 0, \quad y_2(1) = \frac{1}{3}, $

$ \frac{dy_3}{dx} - \frac{(2-x^3)}{x^3} y_3 = 0, \quad y_3(1) = \frac{3}{5e}, $

respectively. Then

$ \lim\limits_{x \to 0^+} \frac{y_1(x)y_2(x)y_3(x) + 2x}{e^{3x} \sin x} $

is equal to __________________.

Let $$f:[1,\infty ) \to [2,\infty )$$ be a differentiable function such that $$f(1) = 2$$. If $$6\int\limits_1^x {f(t)dt = 3xf(x) - {x^3} - 5} $$ for all $$x \ge 1$$, then the value of f(2) is ___________.

Let $f(x)$ be a continuously differentiable function on the interval $(0, \infty)$ such that $f(1)=2$ and

$$ \lim\limits_{t \rightarrow x} \frac{t^{10} f(x)-x^{10} f(t)}{t^9-x^9}=1 $$

for each $x>0$. Then, for all $x>0, f(x)$ is equal to :

Match the statements/expressions in Column I with the values given in Column II:

Match the statements/expressions in Column I with the open intervals in Column II :

STATEMENT-1 : $$y\left( x \right) = \sec \left( {{{\sec }^{ - 1}}x - {\pi \over 6}} \right)$$ and

STATEMENT-2 : $$y\left( x \right)$$ given by $${1 \over y} = {{2\sqrt 3 } \over x} - \sqrt {1 - {1 \over {{x^2}}}} $$

Let $$f:(0,\infty ) \to R$$ be a differentiable function such that $$f'(x) = 2 - {{f(x)} \over x}$$ for all $$x \in (0,\infty )$$ and $$f(1) \ne 1$$. Then

Find the equation of such a curve passing through $$(0,k).$$