The solution of $x \frac{d y}{d x} = y(\log y - \log x + 1)$ is

Let the solution curve of the differential equation $x \frac{dy}{dx} - y = \sqrt{y^2 + 16x^2}$ with the initial condition $y(1) = 3$ be $y = y(x)$. Then the value of $y(2)$ is equal to:

If $P(A') = 0.6$,$P(B) = 0.8$,and $P(B/A) = 0.3$,then $P(A/B) = $

When $3$ dice are thrown at a time, the sum of the numbers appeared on $3$ dice was found to be $15$. Then the probability that the number $5$ does not appear on any one of the dice is

Show that the differential equation $x dy - y dx = \sqrt{x^{2} + y^{2}} dx$ is a homogeneous equation and find its solution.

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 $y = 1$.