Question

The differential equation y" + (sgn x)y = 0 (25) has the discontinuous coefficient function sgnx = S+1 -1 if x > 0, if x < 0. Show that Eq. (25) nevertheless has two linearly indepen- dent solutions yı(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; • Y10) = y0) = 1 and y2(0) = y(0) = 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. y (4) ,6) FIGURE 3.3.2. Graphs of y(x) and y2(x)

Answer 1

- y"+(sgn a) A = 0 – 0 -fx>0) (<037 General sold of43) the c'extes'éx Peneral sols of Treer cos ax + Ca sinx Two linearly independent , Two linearly independent 1 y, - cosmes yetu se (Satisfied) d2 w z = sina) Now let us check yo ! Now let us check yard Satisfy or not 2 to L 1 yt satisfy ③ or not Y," +7,- conx) + Cosse ! 60, *"– 69.) = 2(04). 6°=0 = - cosx #60sx = 0 (Satisfied) (3-60,*) – dr. 6)e-0 y +Y2 = (Sine) + sinx=0! (satisfied). isatisfied) (you= et continuous y = cosx at ao (X) = -e at x=0. y = cosx , y's sinx y* = e, (*) = ex Yo = sinu , ds = cos de l a EX, (*)' = -% 40= 4° 10) = 609031 Cosz éretycy continuous i= -Sink} continuous