The solution of the differential equation $\frac{dy}{dx} - \frac{y+3x}{\log_{e}(y+3x)} + 3 = 0$ is (where $C$ is a constant of integration.)

Find the particular solution of the differential equation $\frac{dx}{dy} = \frac{\sin y(1 + y \cot y)}{x \log(x^2 e)}$,given that $y(1) = 0$.

Find the general solution of the differential equation: $\frac{dy}{dx} = (1+x^2)(1+y^2)$

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

The equation of the curve passing through $(3, 9)$ which satisfies the differential equation $\frac{dy}{dx} = x + \frac{1}{x^2}$ is

Let a curve $y=y(x)$ be given by the solution of the differential equation $\cos \left(\frac{1}{2} \cos ^{-1}\left(e^{-x}\right)\right) d x=\sqrt{e^{2 x}-1} \,d y$. If it intersects the $y$-axis at $y=-1$,and the intersection point of the curve with the $x$-axis is $(\alpha, 0)$,then $e^{\alpha}$ is equal to $.....$