If $\frac{dy}{dx} = f(x, y)$ is a homogeneous differential equation,then the general form of $f(x, y)$ is

Two aeroplanes $I$ and $II$ bomb a target in succession. The probabilities of $I$ and $II$ scoring a hit are $0.3$ and $0.2$,respectively. The second plane will bomb only if the first misses the target. The probability that the target is hit by the second plane is

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 $y = 1$.

Let the solution curve of the differential equation $x dy = (\sqrt{x^{2}+y^{2}}+y) dx$,$x > 0$,intersect the line $x = 1$ at $y = 0$ and the line $x = 2$ at $y = \alpha$. Then the value of $\alpha$ is.

If the general solution of $\frac{dy}{dx} = \frac{y^2}{xy - y^2 - x^2}$ is $\tan^{-1}\left(\frac{y}{x}\right) = f(y) + C$,then $f(e^3) = $

The equation of the curve which passes through point $(1,0)$ and has a tangent with slope $1+\frac{y}{x}+\left(\frac{y}{x}\right)^{2}$ is