1805 Time Invariance

Translation invariance. If $p(D)$ is a constant coefficient differential operator and $p(D)x=q(t)$, then $p(D)y=q(t-c)$, where $y(t)=x(t-c)$.
This is the "time invariance" of p(D). Here is an example of its use.

Example. Suppose that we know that $x_p(t)=\sqrt2\sin (t/2-\pi/4)$ is a solution to the DE $$2x''+x'+x=\sin (t/2)$$ Find a solution $y_p$ to $$2x''+x'+x=\sin (t/2-\pi/3)$$ Solution. By translation invariance, we have immediately that $$y_p=\sqrt2\sin (t/2-\pi/4-\pi/3)=\sqrt2\sin (t/2-7\pi/12)