The particular solution of the differential equation $y(1+\log x) \frac{dx}{dy} - x \log x = 0$ given that $y = e^2$ when $x = e$ is:

The solution of the differential equation $\frac{dx}{dy} + 2yx = 2y$ which passes through the point $(2,0)$ is

The particular solution of the differential equation $\cos \left(\frac{dy}{dx}\right) = 0.5$ with the condition $y = 1$ at $x = 0$ is:

Show that the general solution of the differential equation $\frac{dy}{dx} + \frac{y^{2}+y+1}{x^{2}+x+1} = 0$ is given by $(x+y+1) = A(1-x-y-2xy)$,where $A$ is a parameter.

The particular solution of the differential equation $x \, dy + 2y \, dx = 0$, given that $y = 1$ when $x = 2$, is:

Let $y=y(x)$ be the solution of the differential equation $\log _{e}\left(\frac{d y}{d x}\right)=3 x+4 y$,with $y(0)=0$. If $y\left(-\frac{2}{3} \log _{e} 2\right)=\alpha \log _{e} 2$,then the value of $\alpha$ is equal to: