# How to tell if a system of ordinary differential equations is homogeneous?

Suppose that I have the following set of 2 equations:

$$\frac{d(x(t))}{dt} = 5tx(t) + 2y(t)$$

$$\frac{d(y(t))}{dt} = 5ty(t)+2x(t)$$

I have read that this is a system of "Linear first order non-constant coefficient homogeneous" differential equations. I understand everything except the homogeneous part.

I understand that if you have something like:

$$M(x,y)dx + N(x,y)dy = 0$$ then this equation is homogeneous if $$M$$ and $$N$$ are both homogeneous functions of the same degree but I don't really know how to apply this definition to the set of equations I wrote above.

So, in general, how to tell if a system of ordinary differential equations is homogeneous?

• If you have $$X'(t) = A X(t) + f(t)$$ If $f(t) = 0$, the system is homogeneous, otherwise it is nonhomogeneous. Note that $A$ is $n \times n$ and $f(t)$ are n-vector functions. See: math.upenn.edu/~moose/240S2013/slides7-29.pdf – Moo Jul 15 '20 at 18:48
• It would be really helpful if someone could use this result to prove that the set of equations which I've written are homogeneous. (sorry, as I have never studied n-vector functions) – jon snow Jul 15 '20 at 19:11

In a system of linear differential equations, the only terms allowed are the dependent variables or their derivatives, multiplied by functions of the independent variable, and functions of the independent variable (without any dependent variables). The system is homogeneous if there are no terms without dependent variables or their derivatives. That's the case in your system. If there were other terms such as $$1$$ or $$\sin(t)$$ that don't involve a dependent variable $$x$$ or $$y$$ or its derivative, the system would not be homogeneous.