If y = A sin(x) + B cos(x), then
y' = A cos(x) - B sin(x)
y'' = -A sin(x) - B cos(x)
Substitute these into the ODE:
4y'' + y' + 5y = 2 cos(x)
4 (-A sin(x) - B cos(x)) + (A cos(x) - B sin(x)) + 5(A sin(x) + B cos(x)) = 2 cos(x)
(-4A - B + 5A) sin(x) + (-4B + A + 5B) cos(x) = 2 cos(x)
(A - B) sin(x) + (A + B) cos(x) = 2 cos(x)
Then
A - B = 0
A + B = 2
Adding these equation gives
(A - B) + (A + B) = 0 + 2
2A = 2
A = 1 → B = 1
