Is there any differential equation whose solution is the function itself?

Question

I am not sure if this is a proper question even but would like to know a robust answer, why not. Consider we have a arbitrary function $f(t)$. The variable $t$ can be continuous or discrete. I am looking for a differential equation (DE) (or partial differential equation (PDE) with other variable) whose solution is $f(t)$ itself. In other words the solution of the DE/PDE which is satisfied by the function is the function itself.

An obvious solution to this question is the solution of zero order DE, $y(t)=f(t)$. Is there any other non-zero order DE/PDE which satisfy the above condition?

Edit: After reading the comment of @anomaly I think the question can be written as for what $L$ we have $L(f)=f$.

Answer 1

You set

$$L(y)=y$$ solve for $y$. Then automatically $y=f(t)$ will solve the differential equation.

I will give a simple example $y'=y \implies y(x) = c\exp(x)$. Hence, $y'=c\exp(x)$ is a differential equation of the type that you were looking for.

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Construction of infinitely many differential equations: You can construct infinitely many other examples by starting from $y=g(t)$. Then evaluate some nonlinear $F$ with $F(y,y',...,y^{(n)})$ and set the result of $F$ as your $f(t)$.