If the solution of the differential equation $(2x+3y-2)dx+(4x+6y-7)dy=0$ with $y(0)=3$ is $\alpha x+\beta y+3 \log_e|2x+3y-\gamma|=6$,then $\alpha+2\beta+3\gamma$ is equal to

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

The general solution of $\left(\frac{dy}{dx}\right)^{2} = 1 - x^{2} - y^{2} + x^{2}y^{2}$ is

Let $y=y(x)$ be the solution of the differential equation $dy=e^{\alpha x+y} dx$; $\alpha \in N$. If $y(\log_{e} 2)=\log_{e} 2$ and $y(0)=\log_{e}(\frac{1}{2})$,then the value of $\alpha$ is equal to $.....$

The solution of the equation $(2y - 1) \, dx - (2x + 3) \, dy = 0$ is

If a curve $y = f(x)$ passes through the point $(1, -1)$ and satisfies the differential equation $y(1 + xy)dx = xdy$,then $f(-\frac{1}{2}) = $ . . . . .