The solution of $\frac{dy}{dx} = \frac{ax + b}{cy + d}$ represents a parabola if

Verify that the given function $y = \sqrt{a^{2} - x^{2}}$,where $x \in (-a, a)$,is a solution of the differential equation $x + y \frac{dy}{dx} = 0$ (where $y \neq 0$).

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 equation of the curve passing through $\left(2, \frac{9}{2}\right)$ and having the slope $\left(1-\frac{1}{x^2}\right)$ at $(x, y)$ is

If it is mentioned that for a curve passing through $(3,4)$,the slope of the curve at any point is the reciprocal of twice the ordinate of that point,then that curve is a

Let $f$ be a non-negative function defined in $[0, \pi / 2]$, $f^{\prime}$ exists and is continuous for all $x$, and $\int_0^x \sqrt{1-\left(f^{\prime}(t)\right)^2} dt = \int_0^x f(t) dt$ with $f(0) = 0$. Then