The differential equation

y'' + (sgn x)y = 0 (25)

has the discontinuous coefficient function

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 = 0;
• y₁(0) = y'₂(0) = 1 and y₂(0) = y'₁(0) = 0.

Each y_i(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. 5.3.2.

Transcribed Image Text:

sgn.x = +1 if x > 0, -1 if x < 0.