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Secretion of Hormones 1


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)=αβcos(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).
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(1) In the first step we solve the homogeneous equation, x(t)+kx(t)=0.
Its solutions are xh(t)=Cekt,
where CR 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).


we obtain v(t)=ektαk12βekt[πsin(1/12πt)+12kcos(1/12πt)]π2+144k2.
Hence xp(t)=αk12β[πsin(1/12πt)+12kcos(1/12πt)]π2+144k2.
(3) The general solution of the differential equation is


(4) Since x(0)=x0, we obtain C=144x0k3+144(βα)k2+π2x0kπ2α144k3+π2k,
and x(t)=144x0k3+144(βα)k2+π2x0kπ2α144k3+π2kekt+αk12β[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(βα)k2+π2x0kπ2α144k3+π2kekt+αk12βπ2+144k2sin(π12t+δ),
where tan(δ)=12kπ.
If α,β=1, k=2, and x0=10 then x(t)=e2t(10π22(π2+576))24πsin(1/12πt)+576cos(1/12πt)π25762(π2+576).
Solution by Maple
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