The differential equation $y\frac{dy}{dx} + x = a$ ($a$ is any constant) represents:

The solution of the differential equation $\frac{dy}{dx} + \frac{x}{y} \cdot \frac{x^2+y^2-1}{2(x^2+y^2)+1} = 0$ is

If $y = e^{-mx}$ is a solution of the differential equation $\frac{d^2y}{dx^2} + 4\frac{dy}{dx} + 3y = 0$, then the values of $m$ are

Let $f$ be a differentiable function defined on $\left[0, \frac{\pi}{2}\right]$ such that $f(x) > 0$ and $f(x)+\int \limits_0^x f(t) \sqrt{1-\left(\log _e f(t)\right)^2} d t=e, \forall x \in\left[0, \frac{\pi}{2}\right]$. Then $\left(6 \log _{ e } f \left(\frac{\pi}{6}\right)\right)^2$ is equal to $.............$

$A$ radioactive substance has a half-life of $h \ days$. Its initial decay rate is given by (Note that at $t = 0, M = m_0$):

$A$ body cools according to Newton's law of cooling from $100^{\circ} C$ to $60^{\circ} C$ in $15$ minutes. If the temperature of the surrounding is $20^{\circ} C$,then the temperature of the body after cooling down for one hour is (in $^{\circ} C$)