The general solution of the differential equation $\frac{d^2 y}{d x^2}+8 \frac{d y}{d x}+16 y=0$ is

Find the general solution of the differential equation: $(1+x^{2}) dy + 2xy dx = \cot x dx$ where $x \neq 0$.

At any point on a curve, the slope of the tangent is equal to the sum of the abscissa and the product of the ordinate and abscissa of that point. If the curve passes through $(0, 1)$, then the equation of the curve is

The curve for which the intercept cut off by any tangent on the $y$-axis is proportional to the square of the ordinate of the point of tangency is (where $c_1$ and $c_2$ are arbitrary constants):

If $y=\sin x+A \cos x$ is the general solution of $\frac{dy}{dx}+f(x)y=\sec x$, then an integrating factor of the differential equation is

Let $y=y(x)$ be the solution of the differential equation $\sec^2 x dx + (e^{2y} \tan^2 x + \tan x) dy = 0$,where $0 < x < \frac{\pi}{2}$ and $y(\frac{\pi}{4}) = 0$. If $y(\frac{\pi}{6}) = \alpha$,then $e^{8\alpha}$ is equal to: