The orthogonal trajectories of the family of curves $x^{2/3} + y^{2/3} = a^{2/3}$,where $a$ is an arbitrary constant,are given by:

Number of straight lines which satisfy the differential equation $\frac{dy}{dx} + x \left( \frac{dy}{dx} \right)^2 - y = 0$ is:

Let $\Gamma$ denote a curve $y = y(x)$ which is in the first quadrant and let the point $(1,0)$ lie on it. Let the tangent to $\Gamma$ at a point $P$ intersect the $y$-axis at $Y_p$. If $PY_p$ has length $1$ for each point $P$ on $\Gamma$,then which of the following options is/are correct?
$(1)$ $y=\ln\left(\frac{1+\sqrt{1-x^2}}{x}\right)-\sqrt{1-x^2}$
$(2)$ $xy^{\prime}+\sqrt{1-x^2}=0$
$(3)$ $y=-\ln\left(\frac{1+\sqrt{1-x^2}}{x}\right)+\sqrt{1-x^2}$
$(4)$ $xy^{\prime}-\sqrt{1-x^2}=0$

The population $P = P(t)$ at time $t$ of a certain species follows the differential equation $\frac{dP}{dt} = 0.5P - 450$. If $P(0) = 850$,then the time at which the population becomes zero is:

The rate of increase of bacteria in a certain culture is proportional to the number present. If it doubles in $5$ hours,then in $25$ hours,its number would be ......... times the original.

The rate of disintegration of a radioactive element at time $t$ is proportional to its mass at that time. Then the time during which the original mass of $1.5 \text{ g}$ will disintegrate into its mass of $0.5 \text{ g}$ is proportional to