Problem 1.
The secretion of hormones into the blood is often a periodic activity. If a hormone is secreted on a 24-hour cycle, then the rate of change of the level of the hormone in the blood may be represented by the initial value problem x′(t)=α−βcos(πt12)−kx(t),x(0)=x0, where x(t) is the amount of the hormone in the blood at time t, α is the average secretion rate, β is the amount of daily variation in the secretion, and k is a positive constant reflecting the rate at which the body removes the hormone from the blood.
If α,β=1, k=2, and x0=10, solve for x(t).
(1) In the first step we solve the homogeneous equation, x′(t)+kx(t)=0.
(2) In the second step we solve the inhomogeneous equation, applying the method of variation of parameter(s).
xp(t):=v(t)xh(t).
(4) Since x(0)=x0, we obtain C=144x0k3+144(β−α)k2+π2x0k−π2α144k3+π2k,
Hence x(t)=144x0k3+144(β−α)k2+π2x0k−π2α144k3+π2ke−kt+αk−12β√π2+144k2sin(π12t+δ),
If α,β=1, k=2, and x0=10 then x(t)=e−2t(10−π22(π2+576))−24πsin(1/12πt)+576cos(1/12πt)−π2−5762(π2+576).
Its solutions are xh(t)=Ce−kt,
where C∈R is arbitrary constant.
(2) In the second step we solve the inhomogeneous equation, applying the method of variation of parameter(s).
xp(t):=v(t)xh(t).
x′p(t)=v′(t)xh(t)+v(t)x′h(t),x′p(t)+kxp(t)=v′(t)xh(t)+v(t)[x′h(t)+kxh(t)],=v′(t)xh(t),x′p(t)+kxp(t)=α−βcos(πt12),v′(t)xh(t)=α−βcos(πt12),v′(t)=x−1h(t)[α−βcos(πt12)],v′(t)=ekt[α−βcos(πt12)](C:=1).
After integration by parts
we obtain
v(t)=ektαk−12βekt[πsin(1/12πt)+12kcos(1/12πt)]π2+144k2.
Hence
xp(t)=αk−12β[πsin(1/12πt)+12kcos(1/12πt)]π2+144k2.
(3) The general solution of the differential equation is
x(t)=xh(t)+xp(t),=Ce−kt+αk−12β[πsin(1/12πt)+12kcos(1/12πt)]π2+144k2.
(4) Since x(0)=x0, we obtain C=144x0k3+144(β−α)k2+π2x0k−π2α144k3+π2k,
and
x(t)=144x0k3+144(β−α)k2+π2x0k−π2α144k3+π2ke−kt+αk−12β[πsin(1/12πt)+12kcos(1/12πt)]π2+144k2.
Here we can transform the last term further if we use that
asin(x)+bcos(x)√a2+b2=sin(x+δ),(a,b>0),
where tan(δ)=ba.
Hence x(t)=144x0k3+144(β−α)k2+π2x0k−π2α144k3+π2ke−kt+αk−12β√π2+144k2sin(π12t+δ),
where tan(δ)=12kπ.
If α,β=1, k=2, and x0=10 then x(t)=e−2t(10−π22(π2+576))−24πsin(1/12πt)+576cos(1/12πt)−π2−5762(π2+576).
Solution by Maple