Question

d²y/dx² +4y = e^x +sin2x+cos2x

Answer 1

Answer:

Symbolic form of the given equation D2 + 4y = sin2x Corresponding auxiliary equation D2 + 4 = 0 i.e. D = ±2iThus yC.F = c1cos2x + c2sin2xy P. I. = 1/D2 + 4 sin 2x Replace D2 = -a2 = -4 but here f-a2 = 0[In case of failure 1/fD2 sinax + b = x 1/f-a2sinax + b]Implying y P. I. = x 1/2.D sin2x = x/2-cos2x/2 = xcos2x/4Hence the complete solution y = c1 cos 2x + c2 sin 2x - x cos2x/4