Find the general solution to this differential equation?

Question

d^2y/dx^2 + dy/dx + y = x+1

Answer 1

solve first

D^2 + D + 1 = 0

D = [-1 ± √(-3)]/2

D = -1/2 ± √(3)/2 i

thus

the homogeneous solution is

y = Ae^(-x/2) cos(√(3)x/2) + Be^(-x/2) sin(√(3)x/2)

yp = Ax + B

yp' = A

yp'' = 0

yp'' + yp' + yp = Ax + (A + B) = x + 1

thus

A = 1 , B = 0

yp = x

thus

y = Ae^(-x/2) cos(√(3)x/2) + Be^(-x/2) sin(√(3)x/2) + x