Diff EQ - Heating and cooling w/ fluctuating outside temp

[Indychus]

Homework Statement

Say we have a warehouse with a 5 hr time constant. The outside temp is 16C at 2:00AM, and 32C at 2:00PM. The warehouse is at 24C at noon. What will the temp be at 6:00 pm? When will the temp be 27C?

The Attempt at a Solution

First, I set up a sine wave for the fluctuating outside temp. With x=0 corresponding to 2:00AM, my wave looks like

$$24-8\cos{\frac{2\pi}{24}$$

Here's where it gets fuzzy. From a similar example in class, we set up an equation of form
$$e^\frac{-t}{2}\{\frac{1}{2}\int{e^\frac{t}{2}[M],dt\}+C$$
where $$M$$ is our wave representing the fluctuation of outside temp. The example from class had a time constant of 2 hours, I assume that's where the t/2, -t/2, and 1/2 in the above equation come from?

If I use that equation, but substitute 5 in place of the 2's, then perform the integration, I should be able to differentiate the result and find relative extrema to determine when my temps are at max/min, correct? I should also be able to drop the C (initial temp) term since it will eventually disappear due to exponential decay anyways, right?

 

[lanedance]

so i would interpret the time constant as something like
$$\frac{dT(t)}{dt} = -5(T(t) - T_b)$$

where T_b is the outside temperature, this should be no problem to inetgrat e by separation of variables

rearranging, for the variable outside temp Tb(t) gives
$$\frac{dT(t)}{dt} +5T(t) = 5T_b(t) = 5(24-cos(2 \pi t /24))$$

I would attempt this by trial functions using the method of undetermined coefficients
http://www.efunda.com/math/ode/linearode_undeterminedcoeff.cfm