Population growth (logistic equation)
Population growth can be modeled by
dU(t)
dt
= αU (t),
(15)
where U(t) is the population of a country at time t and α is
the growth rate.
Equation (15) can be re-written as
1
U
dU = α dt
(16)
By integrating equation (16) with respect to each variable, one obtains
(separation of variables)
ln U = αt + C,
(17)
where C is an integral constant.
Equation (17) can be written as
U(t) = eαt + C = eCeαt.
(18)
If the initial population is U0 at t = 0, equation (18) can be
written as
U0 = eC.
(19)
So finally, one obtains
U(t) = U0eαt.
(20)
Correction to the above equation can be made as
dU
dt
= αU (1 −
U
β
)
(21)
where β is known as the carrying capacity.
Analytically solvable differential equations
Analytically solvable differential equations are extremely limited.
The include
Linear differential equations with constant coefficients
Examples:
y"(t) − 3 y′(t) + 2 y(t) = 0, y(0) = a, y′(0) = b.
(Solution): Solve λ2 − 3 λ+ 2 = 0 as λ1 = 1, λ2 = 2.
Then, express y(t) = c1et + c2e2 t. Determine c1 and c2 to
satisfy the initial conditions.
y"(t) + 4 y(t) = 0, y(0) = a, y′(0) = b.
(Solution): Solve λ2 + 4 = 0 as λ1 = 2 i , λ2 = − 2 i.
Then, express y(t) = c1e2 it + c2e− 2 it = c′1 cos 2 t + c′2 sin 2 t. Determine c′1 and c′2 to
satisfy the initial conditions.
Separation of variables
dy
dt
= f(y) g(t) ⇒
1
f(y)
dy = g(t) dt.
Integrating both sides of the equation with respect to each variable, one gets
⌠ ⌡
1
f(y)
dy =
⌠ ⌡
g(t) dt.
Example:
dy
dt
= y (1 − y).
1
y (1 − y)
dy = 1 dt, ⇒
⎛ ⎝
1
y
−
1
y − 1
⎞ ⎠
dy = dt ⇒ (ln y − ln (y − 1) ) = t + C
The general solution is
ln
y
y − 1
= t + C.
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