- #1
- 23,093
- 7,502
Homework Statement
Solve the initial value problem:
##\frac{dx}{dt} = x(2-x)##, ##x(0) = 1##
for ##x(t=ln2)##.
Homework Equations
The Attempt at a Solution
I moved the right side to the left and multiplied both sides by dt to get:
##\frac{dx}{x(2-x)} = dt##
Integrating gave me:
##\frac{ln|x|}{2} - \frac{ln|x-2|}{2} = t+C##
Then:
##ln|x| - ln|x-2| = 2t + 2C##
##ln|\frac{x}{x-2}| = 2t + 2C##
##\frac{x}{x-2} = e^{2t}e^{2c}##
##\frac{1}{1-\frac{2}{x}} = Ke^{2t}##
Manipulating for a while, I end up with:
##x=\frac{2}{Ke^{2t}}##
Since ##x(0) = 1##, I set the left side to 1 and solve for K, winding up with ##k=3##.
However, when trying to solve for ##x(ln2)## I end up with ##\frac{2}{11}##, which isn't one of my possible answers.
Does my process look even remotely correct?