# Clarification of classification of ODE

## Main Question or Discussion Point

Hi there

dr/dz+z^2=0 is classified as a non-homogeneous in Glyn James text, can someone clarify this?

Cheers Petra d.

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One more that I disagree with the text book

dx/dt = f(t)x is classified as homogeneous.

Cheers Petra d.

Samy_A
Homework Helper
dr/dz+z^2=0 is classified as a non-homogeneous in Glyn James text, can someone clarify this?
One more that I disagree with the text book

dx/dt = f(t)x is classified as homogeneous.
At first sight I agree with the textbook.

How do you define a homogeneous ODE, and why do you think the first one is homogeneous and the second one not?

dr/dz+z^2=0

From the text..."right hand side is zero are called homogeneous equations"

dx/dt = f(t)x is classified as homogeneous.

"and those in which it is non-zero are non-homogeneous"

BTW, your avatar looks like you are angry.

Cheers Petra d.

Samy_A
Homework Helper
The term homogeneous ODE can refer to two types of ODE's (see https://en.wikipedia.org/wiki/Homogeneous_differential_equation).

I assume that you are concerned with the second type.

dr/dz+z^2=0

From the text..."right hand side is zero are called homogeneous equations"

dx/dt = f(t)x is classified as homogeneous.

"and those in which it is non-zero are non-homogeneous"
If you use "right hand side" in order to define an homogeneous ODE, you have to be very careful about how you write the ODE.
See, you could write $\frac{dx}{dt}=x'=f(t)x$ as $x'-f(t)x=0$, and now the right hand side is 0.

For a linear ODE, write all the terms that include the unknown function $y$ or its derivatives on the left, and the term that doesn't involve $y$ on the right.
Your two equations (which I will rewrite with $y$ as the function and $t$ as the variable) then become:
$y'=-t²$
$y'-f(t)y=0$.

Do you now see why the textbook is correct?

Alternatively, you could use the definition as given on the Wikipedia page: "A linear differential equation is called homogeneous if the following condition is satisfied: If $\phi(t)$ is a solution, so is $c\phi(t)$, where $c$ is an arbitrary (non-zero) constant. Note that in order for this condition to hold, each term in a linear differential equation of the dependent variable $y$ must contain $y$ or any derivative of $y$. A linear differential equation that fails this condition is called inhomogeneous."

Both methods lead to the same conclusion.
BTW, your avatar looks like you are angry.

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Just so that I understand. You should attempt to have the dep variable on the left and the indep variable on the right before classification?

Cheers Petra

Samy_A