Can these equations be considered as differential equations?

Question

Consider a differential equation with a term containing $y(x_0)$, for example $$y'' - 2y' + y = y(x_0)$$ $x_0 \in \mathbb{R}$ is a constant. My question is, does such equations fall under the category of differential equations? I have never studied any equation with such a term. If its a differential equation, then $y(x_0)$ can be considered a constant coefficient?

Answer 1

Although the term $y(x_0)$ depends on the solution $y$ it is still a constant as it doesn't depend on $x$.

Let's try to solve your equation $y'' - 2y' + y = C,$ where $C=y(x_0).$ One solution is $y(x) = C.$ The solutions of the homogeneous equation are $y(x) = (Ax+B) e^x.$ Therefore the general solutions are $$y(x) = (Ax+B) e^x + C.$$

To get $y(x_0) = C$ we must have $A=B=0.$ Thus there is only the constant solution $$y(x) = C = y(x_0).$$