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 solution of the differential equation $e^x y dx + e^x dy + x dx = 0$ is

Let $S$ be the family of curves given by the general solution of the differential equation $\frac{y^2 e^{-1 / y}}{\sqrt{x}} dx - 2 \sec \sqrt{x} dy = 0$. Then the equation of the curve belonging to $S$ and passing through $(\pi^2, 1)$ is

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

The solution of the differential equation $\frac{dy}{dx} = \cos(x + y)$ is...

The equation of the curve passing through the point $(1, 1)$ such that the slope of the tangent at any point $(x, y)$ is equal to the product of its coordinates is