Can a Nonperiodic Function Solve a Periodic Linear Differential Equation?

[kalish1]

If the forcing function on the right-hand side of a linear nth order differential equation is nonconstant and periodic, can the solution of the equation be a nonperiodic function?

 

[Ackbach]

If the forcing function on the right-hand side of a linear nth order differential equation is nonconstant and periodic, can the solution of the equation be a nonperiodic function?

Sure:
$$\frac{dy}{dx}+y=\sin(x).$$
Solution:
$$y(x)=Ce^{-x}+ \frac{ \cos(x)+ \sin(x)}{2}.$$