- #1

- 24

- 1

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

Cheers Petra d.

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- I
- Thread starter Petra de Ruyter
- Start date

- #1

- 24

- 1

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

Cheers Petra d.

- #2

- 24

- 1

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

Cheers Petra d.

- #3

Samy_A

Science Advisor

Homework Helper

- 1,241

- 510

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

At first sight I agree with the textbook.One more that I disagree with the text book

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

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

- #4

- 24

- 1

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

Samy_A

Science Advisor

Homework Helper

- 1,241

- 510

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.

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.

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.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"

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

Both methods lead to the same conclusion.

BTW, your avatar looks like you are angry.

Last edited:

- #6

- 24

- 1

Cheers Petra

- #7

Samy_A

Science Advisor

Homework Helper

- 1,241

- 510

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

Cheers Petra

Share: