What field should I search to solve delay like equations?

Question

I want to solve equations like: $$ t f(t) + u(t-2) f(t-2) = \sin(t) $$

It's like delay differential equations but without the derivative.

What field should I search in to solve this kind of equation?

Answer 1

Too long for a comment.

You can solve the equation recursively:

1. If $t<2$, $u(t-2)=0$, so the equation becomes $$ tf(t)=\sin(t) \implies f(t)=\frac{\sin(t)}{t}. \tag{1} $$
2. If $2<t<4$, $u(t-2)=1$ and the argument of $f(t-2)$ is less than $2$, so $f(t-2)=\frac{\sin(t-2)}{t-2}$ and $$ f(t)=\frac{1}{t}\left(\sin(t)-\frac{\sin(t-2)}{t-2}\right). \tag{2} $$
3. If $4<t<6$, the argument of $f(t-2)$ is in the interval $(2,4)$, so it is given by $(2)$ with $t$ replaced with $t-2$, hence $$ f(t)=\frac{1}{t}\left(\sin(t)-\frac{1}{t-2}\left(\sin(t-2)-\frac{\sin(t-4)}{t-4}\right)\right). \tag{3} $$

The pattern should be clear by now.