If the population grows at the rate of $5 \%$ per year,then the time taken for the population to become double is (Given $\log 2=0.6912$ ) (in $years$)

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

Find the equation of a curve passing through the point $(0,0)$ and whose differential equation is $y^{\prime}=e^{x} \sin x$.

If the population grows at the rate $5 \%$ per year,then the time taken for the population to become double is $\quad$ (Given $\log 2 = 0.6912$) (in $years$)

Verify that the function $y=c_{1} e^{a x} \cos b x+c_{2} e^{a x} \sin b x,$ where $c_{1}, c_{2}$ are arbitrary constants,is a solution of the differential equation $\frac{d^{2} y}{d x^{2}}-2 a \frac{d y}{d x}+\left(a^{2}+b^{2}\right) y=0$.

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