Verified answer

dy/dx + cos(x)*y = 3*cos(x); y(0) =5.

dy/dx = 3*cos(x) - y*cos(x)

dy/dx = cos(x)(3 - y)

dy/(3 - y) = cos(x) dx

Integrating both sides:

-ln|3 - y| = sin(x) + C

3 - y = C*e^(-sin(x))

y = 3 - C*e^(-sin(x))

5 = 3 - C

C = -2

y = 3 + 2*e^(-sin(x))