Question

微分方程求解答

Answer 1

Answer 2

dx/dt = rx(1- x/xm)
∫dx/[x(xm-x)] = rxm∫ dt
∫dx/[x(x-xm)] = -rxm∫ dt
(1/xm)∫[1/(x-xm) - 1/x ]dx = -rxm.t
(1/xm)ln|(x-xm)/x| =-rxm.t + C'
ln|(x-xm)/x| =-rt + C
x(0) =x0
C = ln|(x0-xm)/x0|
ie
ln|(x-xm)/x| =-rt + ln|(x0-xm)/x0|
x0(x-xm)/[x.(x0-xm)] = e^(-rt)
(x-xm)/x = [(x0-xm)/x0].e^(-rt)
1- xm/x =[(x0-xm)/x0].e^(-rt)
xm/x = 1-[(x0-xm)/x0].e^(-rt)
x = xm/{ 1-[(x0-xm)/x0].e^(-rt) }