The solution of the differential equation $\frac{dy}{dx} = e^{x-y} + x^2e^{-y}$ is...

The solution of the differential equation $2x \frac{dy}{dx} - y = 0$ with the condition $y(1) = 2$ represents . . . . . . .

The particular solution of the differential equation $\frac{dy}{dx} = y \tan x$ with the initial condition $y(0) = 1$ is:

The general solution of the differential equation $\tan(y) dx + \sec^2(y) \tan(x) dy = 0$ 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:

The solution of the differential equation $x \cos y \, dy = (x e^x \log x + e^x) \, dx$ is