The rate of growth of bacteria is proportional to the bacteria present. If it is found that the number doubles in $3$ hours,then the number of times the bacteria are increased in $6$ hours is

The number of solutions of $y' = \frac{y + 1}{x - 1}, y(1) = 2$ is

If the equation of the curve which passes through the point $(1,1)$ satisfies the differential equation $\frac{dy}{dx} = \frac{2x-5y+3}{5x+2y-3}$, then the equation of that curve is:

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

Let $f:[0,2] \rightarrow R$ be a function which is continuous on $[0,2]$ and is differentiable on $(0,2)$ with $f(0)=1$. Let $F(x)=\int_0^{x^2} f(\sqrt{t}) dt$ for $x \in [0,2]$. If $F'(x)=f'(x)$ for all $x \in (0,2)$,then $F(2)$ equals

The rate of growth of bacteria in a culture is proportional to the number of bacteria present and the bacteria count is $1000$ at initial time $t = 0$. The number of bacteria is increased by $20\%$ in $2$ hours. If the population of bacteria is $2000$ after $\frac{k}{\log_{e}\left(\frac{6}{5}\right)}$ hours,then $\left(\frac{k}{\log_{e} 2}\right)^{2}$ is equal to