Take the Verhulst logistic differential equation down blew:
$$I'(t) = \beta I(t)\left(1-\dfrac {I(t)}N \right)$$
with beta an N as constants.
How can I write this equation as a f(t,y) equation int the form of y'(t)=f(t,y(t)) (here y(t)=I(t))?
I want to write in this function so I can calculate partial derivative of it and see the existence and uniqueness of the solution.
Thanks in advance.
$$I'(t) = \beta I(t)\left(1-\dfrac {I(t)}N \right)$$ $$I'(t) = \beta \left(I(t)-\dfrac {I^2(t)}N \right)$$ It's alreay in the form: $$y'=f(t,y)$$ Here $y(t)=I(t)$
For the derivative with respect to $I$ $$\dfrac {\partial }{\partial I} \beta \left(I(t)-\dfrac {I^2(t)}N \right)= \beta \left(1-\dfrac {2I(t)}N \right)$$