Find the general solution of the logistic differential equation:
Using slope field analysis (and possibly a computer generated slope field) explain why you are confident that your family of solutions is correct?
variables are separable.
where c is constant of integration.
C can be determined by given condition.
`dy/dx=` `y(2-y)` `(dy)/(y(2-y))=dx`
integrating both sides:
`=1/2(1/y +1/(2-y)) dy= dx`
`log[sqrt(y(2-y))] = x`
`2cy -cy^2 - e^(2x)= 0`
rewriting in a fit shape:
``adding and subtracting c:
`cy^2-2cy+c -c+e^(2x) =0`
so the function we'r searching for is:
this funcion is definite for `c` `!= 0` , since c depends by a initial condition it doesn't affetc define field of the function .
Once we have assigned c, can lok at field, as the argument inside root is to be greater equal zero.
since the function `e^(2x)` increasing it means : `x<|log c|`
from arbitray choice of c,(however it does suggest us c>0)
Now we calculate slope: