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

The particular solution of the differential equation $y(1+\log x) \frac{dx}{dy} - x \log x = 0$ given that $y = e^2$ when $x = e$ is:

The curve passing through the point $(1,2)$ given that the slope of the tangent at any point $(x, y)$ is $\frac{3x}{y}$ represents

The general solution of the differential equation $\cos x \sin y \, dx + \sin x \cos y \, dy = 0$ is

If $y = y(x)$ is the solution of the differential equation $(1 + e^{2x}) \frac{dy}{dx} + 2(1 + y^2)e^x = 0$ and $y(0) = 0$,then $6(y'(0) + (y(\log_e \sqrt{3}))^2)$ is equal to

The solution of the differential equation $x(e^{2y} - 1)dy + (x^2 - 1)e^y dx = 0$ is