Question $2$: $48$ Marks

Find the general solutions $to$ the following differential equations:

$(2\mathrm{.}1)$

$(tln(x))\frac{dx}{dt}=(\frac{t+1}{x}{)}^2$

$(2\mathrm{.}2)$

$(e^x+1{)}^3{e}^{-x}\frac{dy}{dx}=-(e^y+1{)}^2{e}^{-y}$

$28$

$(2\mathrm{.}3)$

$\frac{dy}{dx}=\frac{2y-x-2+xy}{xy+x-3-3y}$

$\frac{dN}{dt}=t\sqrt[\phantom{2}]{1-N^2}$

$\frac{dN}{dt}=\frac{-t^3}{(N+1{)}^2}$