Clarification of classification of ODE

In summary: If you use the definitions found on the Wikipedia page, no.The classification of ODEs is not a particularly interesting or important topic, so I suggest that we stop here.
  • #1
Petra de Ruyter
24
1
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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  • #2
One more that I disagree with the textbook

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

Cheers Petra d.
 
  • #3
Petra de Ruyter said:
dr/dz+z^2=0 is classified as a non-homogeneous in Glyn James text, can someone clarify this?
Petra de Ruyter said:
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?
 
  • #4
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
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.

Petra de Ruyter said:
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.
Petra de Ruyter said:
BTW, your avatar looks like you are angry.
:oldsmile:
 
Last edited:
  • #6
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
Petra de Ruyter said:
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.
 

What is an ODE?

An ODE, or ordinary differential equation, is a mathematical equation that describes the relationship between a function and its derivatives. It typically involves a function and one or more of its derivatives with respect to one or more independent variables.

What are the different types of ODEs?

There are several types of ODEs, including first-order, second-order, and higher-order. First-order ODEs involve only the first derivative of the function, while second-order ODEs involve the second derivative. Higher-order ODEs involve derivatives of even higher orders.

How are ODEs classified?

ODEs can be classified based on several factors, including the order of the equation, the number of independent variables, and the linearity of the equation. They can also be classified as initial value problems or boundary value problems.

What is the difference between an initial value problem and a boundary value problem?

In an initial value problem, the values of the function and its derivatives are known at a specific point, and the goal is to find the function that satisfies the equation at that point. In a boundary value problem, the values of the function are known at multiple points, and the goal is to find the function that satisfies the equation at all of those points.

How are ODEs solved?

There are various methods for solving ODEs, including analytical methods, numerical methods, and computer simulations. The choice of method depends on the type and complexity of the equation, as well as the desired level of accuracy.

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