The differential equation y00 C .sgn x/y D 0 (25) has the
Chapter , Problem 50(choose chapter or problem)
The differential equation y00 C .sgn x/y D 0 (25) has the discontinuous coefficient function sgn x D ( C1 if x>0, 1 if x<0. Show that Eq. (25) nevertheless has two linearly independent solutions y1.x/ and y2.x/ defined for all x such that Each satisfies Eq. (25) at each point x 0; Each has a continuous derivative at x D 0; y1.0/ D y0 2.0/ D 1 and y2.0/ D y0 1.0/ D 0. (Suggestion: Each yi.x/ will be defined by one formula for x<0 and by another for x = 0.) The graphs of these two solutions are shown in Fig. 3.3.2. 2
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