$A$ die is thrown twice. If the sum of the numbers obtained is $6$,what is the probability that the number $4$ appears at least once?

Let $f(x) = \sqrt{\lim_{r \rightarrow x} \left\{ \frac{2r^2 \left[(f(r))^2 - f(x)f(r)\right]}{r^2 - x^2} - r^3 e^{\frac{f(r)}{r}} \right\}}$ be differentiable in $(-\infty, 0) \cup (0, \infty)$ and $f(1) = 1$. Then the value of $ea$,such that $f(a) = 0$,is equal to:

The solution of the differential equation $y \frac{dy}{dx} = x \left[ \frac{y^2}{x^2} + \frac{\phi(y^2/x^2)}{\phi'(y^2/x^2)} \right]$ is (where $c$ is a constant):

Find the particular solution satisfying the given condition:
$\frac{dy}{dx} - \frac{y}{x} + \csc\left(\frac{y}{x}\right) = 0$; $y = 0$ when $x = 1$.

Show that the differential equation $2 y e^{\frac{x}{y}} dx + (y - 2 x e^{\frac{x}{y}}) dy = 0$ is homogeneous and find its particular solution,given that $x = 0$ when $y = 1$.

The particular solution of the differential equation,$x y \frac{dy}{dx} = x^2 + 2y^2$ when $y(1) = 0$ is