Let y = y(x) be the solution of the differential equation dy = e^{$$\alpha$$x + y} dx; $$\alpha$$ $$\in$$ N. If y(log_e2) = log_e2 and y(0) = log_e$$\left( {{1 \over 2}} \right)$$, then the value of $$\alpha$$ is equal to _____________.

If $$y = y(x),y \in \left[ {0,{\pi \over 2}} \right)$$ is the solution of the differential equation $$\sec y{{dy} \over {dx}} - \sin (x + y) - \sin (x - y) = 0$$, with y(0) = 0, then $$5y'\left( {{\pi \over 2}} \right)$$ is equal to ______________.

Let a curve y = f(x) pass through the point (2, (log_e2)²) and have slope $${{2y} \over {x{{\log }_e}x}}$$ for all positive real value of x. Then the value of f(e) is equal to ______________.

Let y = y(x) be solution of the following differential equation $${e^y}{{dy} \over {dx}} - 2{e^y}\sin x + \sin x{\cos ^2}x = 0,y\left( {{\pi \over 2}} \right) = 0$$ If $$y(0) = {\log _e}(\alpha + \beta {e^{ - 2}})$$, then $$4(\alpha + \beta )$$ is equal to ______________.