We have (D2 + 1)y = tanx
A.E. is m2 + 1 = 0
i.e., m2 = – 1
i.e., m = ± i
C.F. is C.F. = yc = C1cos x + C2sinx
∴ y = A cos x + B sinx ...(1)
be the complete solution of the given equation where A and B are to be found
We have y1 = cos x and y2 = sin x
y′1 = – sinx y′2 = cos x
W = y1y'2 - y2y'1
= cos x . cos x + sin x . sin x = cos2x + sin2x = 1
Substitute these values of A and B in Eqn. (1), we get
y = {– log(secx + tanx) + sinx + C1} cosx + {– cos x + C2} sinx
y=C1cos x + C2sin x – cosx log (sec x + tan x)
This is the complete solution.
