The equation to the orthogonal trajectories of the system of parabolas $y = ax^2$ is

The population of a city increases at a rate proportional to the population at that time. If the population of the city increases from $20$ lakhs to $40$ lakhs in $30$ years,then after another $15$ years,the population is:

Let $y = y(x)$ be the solution of the differential equation $x dy - y dx = \sqrt{x^2 - y^2} dx$,$x \geq 1$,with $y(1) = 0$. If the area bounded by the lines $x = 1$,$x = e^{\pi}$,$y = 0$ and the curve $y = y(x)$ is $\alpha e^{2\pi} + \beta$,then the value of $10(\alpha + \beta)$ is equal to ....... .

Let $\int_0^x \sqrt{1-\left(y^{\prime}(t)\right)^2} dt = \int_0^x y(t) dt, 0 \leq x \leq 3, y \geq 0$,$y(0)=0$. Then at $x=2, y^{\prime \prime}+y+1$ is equal to :

Let $f$ be a twice differentiable non-negative function such that $(f(x))^2 = 25 + \int_{0}^{x} ((f(t))^2 + (f'(t))^2) dt$. Then the mean of $f(\log_e(1)), f(\log_e(2)), \ldots, f(\log_e(625))$ is equal to:

The rate at which the population of a city increases varies as the population. In a period of $20$ years,the population increased from $4$ lakhs to $6$ lakhs. In another $20$ years,the population will be (in $lakhs$)