How do I determine a homogenous differential equation?

Question

I have been given a system of differential equations but I'm not quite sure how to determine whether it is homogeneous or non homogeneous.

The system is:

dx/dt= 8sin(t)x + ln(t)y;

dy/dt= (4t^2)y + (2e^t)x

What steps would I take to determine whether it is homogeneous or not?

Answer 1

The other answers have given you the shortcut. It's good to know, but shortcuts are hard to use and/or remember if you don't know what's going on in the first place.

Let's review some facts:

1. Remember from the beginning of the course, you learned that a linear equation y'+Py = f is homogeneous if f=0 (the term comes from linear algebra).
2. Remember that P and f are allowed to be functions. So the full written version is y'(x) + P(x)y(x) = f(x)
3. You can have more derivatives, but a linear equation always looks like a_ny⁽ⁿ⁾+...+a₂y''+a₁y'+ay=f
4. The term "term" only refers to things being added together. If you have 4xsinx+8abcx², 4xsinx is a term, sinx is not, abc is not. This is important for understanding why the other answers to your post are correct.
5. The system you gave is also first order linear and the idea is it can look the same if you use a matrix: X'+AX = f (with your variables X=(x,y)
6. Still functions: X'(t)+A(t)X(t) = f(t)
7. I wrote A on the left to look like the y'+Py, but if you write X'=AX+f, then the sign will be switched!

So the other two answers are assuming you know the equation is linear and first order, and that you understand what that means. The equation type will only allow dependent variables to show up like [t stuff]*x or [t stuff]*y. So either a term has an x or y and looks like those, or it's part of f(t).

Answer 2

How to Tell if a System of Differential Equations Is Homogeneous

Imagine you’re walking into the land of differential equations, and someone hands you this pair of mysterious creatures:

dx/dt = 8*sin(t)*x + ln(t)*y

dy/dt = 4*t^2*y + 2*e^t*x

You’re told they form a system — two variables, x and y, both evolving with time t.

But the question is: are they homogeneous or not?

Setting the Scene

In the world of linear systems, every creature (each differential equation) can be described in one neat matrix sentence:

X' = A(t)*X + b(t)

where

1. X is a column matrix whose entries are the evolving quantities x and y,
2. A(t) is the system’s personality matrix, telling how x and y interact, and
3. b(t) is an external forcing term — like an outside influence that pushes the system whether or not x or y are there. Note that b(t) is a column matrix as well.

The Heart of the Definition

Now, here’s the key idea — the moral of the story:

If the system has no external push — that is, if b(t) = 0 everywhere — the system is called homogeneous.

But if there is such a push, some f(t), g(t), or any term that doesn’t multiply x or y, then the system becomes non-homogeneous.

It’s like the difference between a boat drifting because of its own currents (homogeneous), versus being tugged by an outside rope (non-homogeneous).

Applying the Rule

Look again at your system.

Every term on the right-hand side — (8sin(t)x), (ln(t)y), (4t^2y), (2e^t*x) — is attached to either x or y.

No lonely term like “+sin(t)” or “+e^t” appears by itself.

That means there’s no external forcing vector b(t).

So, in matrix form:

X' = [ [8*sin(t), ln(t)],

[2*e^t, 4*t^2] ] * X

and b(t) = 0.

The Verdict

Since b(t) is zero, your system is homogeneous.

A Broader Intuition — The Lotka–Volterra Example

Now, let’s step out of pure symbols and into the real world for a moment.

Imagine an ecosystem — a small island where rabbits and foxes live together. The rabbits (x) multiply when food is plenty, and the foxes (y) thrive when there are enough rabbits to hunt. Their story unfolds through this pair of equations:

dx/dt = a*x - b*x*y

dy/dt = d*x*y - g*y

Here, the first equation says the rabbit population grows on its own (the ax part) but drops when foxes catch them (the -bxy part). The second says the fox population depends on how many rabbits are around (the dxy part) but declines naturally when food is scarce (the -gy part).

Everything that happens here comes from inside the system — from the way rabbits and foxes affect each other. There’s no outside interference, no extra food shipments, no hunters arriving. It’s a perfectly homogeneous world — a closed system driven entirely by its own internal interactions. But now imagine that, every spring, humans drop extra food for the rabbits. That small seasonal help can be written as an extra term added to the first equation:

dx/dt = a*x - b*x*y + f(t)

That new friend, f(t), is an external push — a source of energy or resources coming from the outside. And just like that, the peaceful island is no longer isolated. The system has become non-homogeneous, because the change in x now depends not just on the rabbits and foxes, but also on what’s happening beyond their little world.

Answer 3

To check if a system is homogeneous:

Look for terms that don’t involve the dependent variables.

If none exist (i.e., every term is a function of and/or ), then the system is homogeneous.

Look at the right-hand sides:

Each term involves either or , and there are no standalone terms like just , , or constants.

The system is homogeneous because every term on the right-hand side is multiplied by or —there are no external forcing functions.

Answer 4

It is homogenous if every term involves a dependent variable (x, y) or its derivative (dx/dt, dy/dt).

And it is so in this case -- there is no term involving the independent variable t alone.