Calculating anything without initial data is veeeeeery troublesome =)

du/dt + 1/2*(1+u)^2 = 0

du/dt+ 1/2*(1^2+2u+u^2) = 0

du/dt +1/2 + u + u^2 = 0

du/dt + u = -u^2 - 1/2 this gives you ke^-x

Set

gives you uŽ= 2ax +b

gives you a = -1

b = 2

gives you c = -2,5

To determine K, I think you need a value somewhere,

If you plot that graph for different values of K you will see that the slopes have different angles exept for the x value for maximum/minimum point, is that the value you are trying to find? Otherwise you need some initial data to determine K. ( I think)

hope it was somewhat helpful.