If it is mentioned that for a curve passing through $(3,4)$,the slope of the curve at any point is the reciprocal of twice the ordinate of that point,then that curve is a

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 assets of a person are reduced in his business such that the rate of reduction is proportional to the square root of the existing assets. If the assets were initially ₹ $10,00,000$ and due to loss they reduce to ₹ $10,000$ after $3$ years,then the number of years required for the person to go bankrupt will be

If a curve passes through the origin,such that the length of the subnormal is equal to one more than the square of the ordinate,then:

The rate of decay of mass of a certain substance at time $t$ is proportional to the mass at that instant. The time during which the original mass of $m_{0}$ grams will be reduced to $m_{1}$ grams is (where $k$ is the constant of proportionality):

The solution of the differential equation $x \frac{d^2y}{dx^2} = 1$,given that $y = 1$ and $\frac{dy}{dx} = 0$ when $x = 1$,is