# I Clarification of classification of ODE

1. Apr 13, 2016

### Petra de Ruyter

Hi there

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

Cheers Petra d.

2. Apr 13, 2016

### Petra de Ruyter

One more that I disagree with the text book

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

Cheers Petra d.

3. Apr 13, 2016

### Samy_A

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?

4. Apr 13, 2016

### Petra de Ruyter

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.

5. Apr 14, 2016

### Samy_A

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.

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.

Last edited: Apr 14, 2016
6. Apr 14, 2016

### Petra de Ruyter

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

7. Apr 15, 2016

### Samy_A

If you use concepts like "right hand side" in your classification, yes.