The general solution of the differential equation $\frac{dy}{dx} = e^{y+x} + e^{y-x}$ is, where $c$ is an arbitrary constant.

If a curve passes through the point $(1,1)$ and at any point $(x, y)$ on the curve,the product of the slope of its tangent and $x$ coordinate of the point is equal to the $y$ coordinate of the point,then the curve also passes through the point

The general solution of the differential equation $\frac{dy}{dx} = 1 - x + y - xy$ is (where $C$ is a constant of integration)

If the solution $y=y(x)$ of the differential equation $(x^4+2x^3+3x^2+2x+2)dy-(2x^2+2x+3)dx=0$ satisfies $y(-1)=-\frac{\pi}{4}$,then $y(0)$ is equal to:

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

If the curve $y = f(x)$ passes through the point $(1, e)$ and satisfies the differential equation $dy = y(2 + \log_e x) dx, x > 0$, then $f(e)$ is equal to: