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 the solution of the differential equation ${{dy} \over {dx}} = 2(y + 2\sin x - 5)x - 2\cos x$ such that y(0) = 7. Then…

Let $x=x(y)$ be the solution of the differential equation $2(y+2) \log _{e}(y+2) d x+\left(x+4-2 \log _{e}(y+2)\right) d y=0, y-1$ with…

If $${{dy} \over {dx}} = {{{2^x}y + {2^y}{{.2}^x}} \over {{2^x} + {2^{x + y}}{{\log }_e}2}}$$, y(0) = 0, then for y = 1, the value of x…

Let y = y(x) be the solution of the differential equation $x(1 - {x^2}){{dy} \over {dx}} + (3{x^2}y - y - 4{x^3}) = 0$, $x 1$, with $y(2) =…

Let $y=y(x)$ be the solution curve of the differential equation $$\sin \left( {2{x^2}} \right){\log _e}\left( {\tan {x^2}} \right)dy +…