Q. What do I do to solve

                u'+ku=k a(t)?

A. Use a pencil and paper. Apply the linear integrating factor method
   from section 1.5 of the textbook [suggestion by Y. Chen], which applies
   to linear equations du/dt+p(t)u=q(t). The heat equation u'+ku=k a(t) is
   a linear first order differential equation with p(t)=k and q(t)=k a(t).

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Q. What is the superposition principle y = y_h + y_p?

A.  The non-homogeneous equation is
   
                u'+ku=k a(t)

   and the homogeneous equation is

                u'+ku=0.

   Their solutions are abbreviated u_p and u_h, respectively.
   Superposition says u=u_h+u_p. The expression for u_h contains a
   constant C (or an equivalent symbol u(0) or u_0). The expression
   u_p has no such symbols.
   
   From Growth-decay theory for y'=ay, solution y=C exp(ax), it follows
   that u_h(t)=C exp(-kt). The symbol C is an arbitrary constant, to be
   resolved later from initial conditions.

   The integrating factor method is used to find a formula for u_p.
   Normally, this involves replacing the left side to get

                (Wu)'/W = k a(t), W=exp(k t)                                
 
   The answer for u_p is a definite integral as in the lab notes,
   because the lower limit 0 in the integral allows the resolution of C
   as the constant u(0)=wall thermometer reading at t=0.
   
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Q. How do I integrate a(t)?

A. Use maple and attach a print to the end of L2.1.
   You are not expected to do it by hand, even though you
   probably once owned that talent.


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Q. How do I check the answer?

A. Use maple as per the instructions in lab 2. Attach a print of
   the maple worksheet to the end of L2.1.

   If your check does not get zero as the evaluation, then look for an
   undefined symbol like pi (Pi is correct, upper case P and lower case
   i). 

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