The solution of $\frac{d^2 y}{d x^2}=0$ represents

The population of a town increases at a rate proportional to the population at that time. If the population increases from $40,000$ to $80,000$ in $20$ years,then the population in another $40$ years will be (in $,000$)

$A$ curve passes through the point $(3,2)$ for which the segment of the tangent line contained between the coordinate axes is bisected at the point of contact. The equation of the curve is

The slope of the tangent to a curve $C : y = y(x)$ at any point $(x, y)$ on it is $\frac{2e^{2x} - 6e^{-x} + 9}{2 + 9e^{-2x}}$. If $C$ passes through the points $(0, \frac{1}{2} + \frac{\pi}{2\sqrt{2}})$ and $(\alpha, \frac{1}{2}e^{2\alpha})$,then $e^{\alpha}$ is equal to:

Radium decomposes at a rate proportional to the amount present. If half the original amount disappears in $1600$ years,then the percentage loss in $100$ years is (Given $\log 2 = 0.6931$ and $e^{-0.0433} = 0.9576$) (in $\%$)

Verify that the given function $y=x \sin 3x$ is a solution of the differential equation $\frac{d^{2}y}{dx^{2}}+9y-6 \cos 3x=0$.