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

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 3

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.