46. Assume that all functions in this exercise are defined on a common interval .a; b/.
(a) Prove: If y1 and y2 are solutions of
y0 C p.x/y D f1.x/
and
y0 C p.x/y D f2.x/
respectively, and c1 and c2 are constants, then y D c1y1 C c2y2 is a solution of
y0 C p.x/y D c1f1.x/ C c2f2.x/: (This is theprinciple of superposition.)
(b) Use (a) to show that if y1 and y2 are solutions of the nonhomogeneous equation
y0 C p.x/y D f .x/; .A/
then y1 � y2 is a solution of the homogeneous equation
y0 C p.x/y D 0: .B/
(c) Use(a)toshowthatify1 isasolutionof(A)andy2 isasolutionof(B),theny1 Cy2 isa solution of (A).