The integrating factor of the differential equation $x \frac{dy}{dx} + 2y = x^2 \log x$ is

Let $x = x(y)$ be the solution of the differential equation $2(y + 2) \log_e(y + 2) dx + (x + 4 - 2 \log_e(y + 2)) dy = 0$,$y > -1$ with $x(e^4 - 2) = 1$. Then $x(e^9 - 2)$ is equal to

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

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

If $y=y(x)$ is the solution of the differential equation $e^{y}\left(\frac{dy}{dx}-1\right)=e^{x}$ such that $y(0)=0,$ then $y(1)$ is equal to

The equation of the curve passing through the point $(0,2)$ given that the sum of the ordinate and abscissa of any point exceeds the slope of the tangent to the curve at that point by $5$ is