The general solution of the equation $\frac{dy}{dx} + \frac{1}{x}y = \frac{1}{x}e^x$ is

The solution of the differential equation $x \frac{dy}{dx} + y = x^3 y^6$ is:

If the solution curve $f(x, y)=0$ of the differential equation $(1+\log_e x) \frac{dx}{dy} - x \log_e x = e^y, x > 0$,passes through the points $(1,0)$ and $(\alpha, 2)$,then $\alpha^\alpha$ is equal to

Let $y=y(x)$ be the solution of the differential equation $(\tan x)^{1/2} dy = (\sec^3 x - (\tan x)^{3/2} y) dx$, where $0 < x < \frac{\pi}{2}$ and $y(\frac{\pi}{4}) = \frac{6\sqrt{2}}{5}$. If $y(\frac{\pi}{3}) = \frac{4}{5}\alpha$, then $\alpha^4$ equals . . . . . . .

The general solution of $\frac{dy}{dx} + y \tan x = 2x + x^2 \tan x$ is:

If $\cos x \frac{dy}{dx} - y \sin x = 6x$,where $0 < x < \frac{\pi}{2}$ and $y(\frac{\pi}{3}) = 0$,then find $y(\frac{\pi}{6})$.