Clarification of classification of ODE

[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.

 

[Petra de Ruyter]

One more that I disagree with the textbook

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

Cheers Petra d.

 

[Samy_A]

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 textbook

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?

 

[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.

 

[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.

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.

 

[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

 

[Samy_A]

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

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