EXERCISES 2.2
2.2.1. Show that any linear combination of linear operators is a linear operator.

2.2.2. (a) Show that L(u) = (d/dx)(K0(x) du/dx) is a linear operator.
       (b) Show that usually L(u) = (d/dx)(K0(x,u) du/dx) is not a linear operator.

2.2.3. Show that du/dt = k(d/dx)^2 u + Q(u,x,t) is linear if Q=alpha(x,t) u + beta(x,t) and, in
addition, homogeneous if beta(x,t)=0.

2.2.4. In this exercise we derive superposition principles for nonhomogeneous problems.
(a) Consider L(u) = f. If up is a particular solution, L(up) = f, and
if ul and u2 are homogeneous solutions, L(u1) = L(u2) = 0, then show that u =
up + c1 u1 + c2 u2 is another particular solution, that is, show that L(u)=f.
(b) If L(u1)=f1 and L(u2)=f2, then
what is a particular solution u for L(u) = f1 + f2?

2.2.5 If L is a linear operator and L(u1)=L(u2)=L(u3)=0, then show that 
L(c1 u1 + c2 u2 + c3 u3) = 0. Use this result to show that the principle of superposition 
may be extended to any finite number of homogeneous solutions.