Solve the initial value problem:
dx/dt = x(2-x) x(0) = 1
The Attempt at a Solution
Based on the format, I attempted to solve the problem as a separable differential equation:
∫dx/(x[2-x]) = ∫dt
(ln|x|)/2 - (ln|2 - x|)/2 + C = t + C
ln|x| - ln|2 - x| + 2C = 2t + 2C
Cancelling the constant
ln|x| - ln|2 - x| = 2t
Removing the logs
x - (2 - x) = e^2t
2x - 2 = e^2t
And finally solving for x:
(e^2t)/2 + 1
Which gives me a function, but for t = 0, x(0) = 3/2, not 1.
Should I be using another technique, or did I make a mistake somewhere in the process above?