# Solving first order “differential equations” with different inputs for the function and derivative?

For the differential equation

$dy/dx = ky$

there exists the solution:

$y = y_{0}e^{k(x-x_0)}$

However, when we compute this solution, we do so under the implicit assumption that $y$ is a function of $x$. We could rewrite the differential equation as

$d(y(x))/dx = ky(x)$

and it would mean the exact same thing.

My question: is there an area of mathematics dedicated to studying "differential equations" of the form

$d(y(x + a))/dx = ky(x)$

where $a$ is an arbitrary constant? Furthermore, if such an area of mathematics does exist, what level of math education would one need to understand it?

I'm not even sure what to call this kind of equation, hence my use of quotation marks around "differential equation." My intuition tells me that there are a truckload of different potential solutions to this equation, but I have no idea how one would go about computing them.

• If I understand you right: When $a$ is a constant then $f(x+a)$ is just a function of $x$. E.g. : $f(x)=\sqrt{x+a}$. That's just a function of x alone. – Rutger Moody Mar 5 '17 at 21:35
• @Rutger That's true, but it can't easily be related to $f(x)$ in most cases, which makes the equation harder to solve. This is a functional equation. en.m.wikipedia.org/wiki/Functional_equation – Matt Samuel Mar 5 '17 at 21:43
• Try looking at Laplace transform – N74 Mar 5 '17 at 21:43
• What you're writing is called a delay differential equation or DDE, and would more commonly be written as $\frac{dy}{dx}=ky(x-a)$. They are closer to PDEs than to ODEs in their difficulty to understand, because in effect the value of the solution at a given value of $x$ is really a function defined on $[x-a,x]$. In this respect they are somewhat like an evolution PDE like the heat equation or the like. – Ian Mar 5 '17 at 22:21