The solution of the differential equation $y \frac{dy}{dx} + x = k$ represents . . . . . . .

The curve passing through $(3, 0)$ and satisfying the differential equation $(9 - x^2)(\frac{dy}{dx})^2 = 9 - y^2$ represents:

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

The bacteria increase at a rate proportional to the number of bacteria present. If the original number $N$ doubles in $8$ hours,then the number of bacteria in $24$ hours will be (in $N$)

The rate of increase of the population of a city is proportional to the population present at that instant. In the period of $40$ years,the population increased from $30,000$ to $40,000$. At any time $t$,the population is given by $P(t) = (a)(b)^{\frac{t}{40}}$. Then the values of $a$ and $b$ are respectively: