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). ======================================================================== 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. ======================================================================== 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. ======================================================================== 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). ======================================================================== Q. How do I determine the inequalities to report in L2.4? A. Print the 3D graphic and the 2D graphic found in the problem notes of maple Lab2. They are found at the end of the lab. You will see on the 2D graphic the intersection of the plane z=30F with the temperature surface z=z(t,k). In the first 40 hours, the surface drops below z=30F for (t,k)-values trapped between two curves. This defines a region R in the (t,k)-plane described by inequalities, similar to regions used in double integration problems: R = { (t,k) : a < k < b, T_1(k) < t < T_2(k) } The approximate answer fits a rectangle R_1 inside R, using the maple graphic that displayed the two mystery curves. Then R is replaced by R_1 = { (t,k) : a < k < b, c < t < d } Expect to report the inequalities for R_1, for three such regions R. No matter how unusual the curves might appear, simply fit a rectangle inside the region between the curves, and report the inqualities that describe the rectangle.