Please Help Outside a building, the temperature is T0 (degree C) and does not ch

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Outside a building, the temperature is T0 (degree C) and does not change. The building is heated during a fraction lambda of on hour, Starting at the beginning of every hour. There are no other sources of heating or cooling in the building. Assume that the heater generate the constant amount h thousands Btu/hr when it is on, the heat capacity of the building is eta degree per thousand Btu, and the time constant for heat transfer between the outside and the inside of building is tau hr. Solve the IVP: dx/dt = T0 - x/tau + eta H(t), x(0) = T*, for the time interval 0 le t le 7/4 hr, where T0 = 0 , lambda = 3/4, h = 6, eta = 1/2, tau = 6, T* = 18. Compare the temperature in the building in 1/2 hr to T*. Directions

Explanation / Answer

so dx/dt = -x/6 +1/2 * H

H = h*u(t) - h*u(t-3/4) + h*u(t-1) - h*u(t-3/4)

so

dx/dt = -x/6 + 3*( u(t) - u(t-3/4) + u(t-1) - u(t-7/4))

laplace transform

dx/dt -> sX - 18



u(t-c) = e^(-cs)/s

so becomes

sX - 18 =-X/6 + 3*(1/s - e^(-3/4s)/s + e^(-s)/s - e^(-7/4s)/s)

X = (18+3*(1/s - e^(-3/4s)/s + e^(-s)/s - e^(-7/4s)/s))/(s+1/6)

transform back

x=18 ((e^(7/24-t/6)-1) u(t-7/4)+(1-e^(1/6-t/6)) u(t-1)+e^(1/8-t/6) u(t-3/4)-u(t-3/4)+1)

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