Notation for separable partial differential equations

In summary: Ux=X'Y, and Uy=X''Y, etc.In summary, the notation used for partial derivatives in the context of separation of variables can be a bit confusing. Uxx typically means the second order partial derivative of U with respect to x, while Ux can mean either the first order partial derivative of U with respect to x or the derivative of the partial derivative of U with respect to x. It is important to clarify the meaning of U'x in this context to avoid confusion.
  • #1
frozenguy
192
0
Hey.. I ran across some problems and the notation used is a little different from what I've seen before.

considering U(x,y)=X(x)Y(y)
Sometimes I'll see Uxx for [tex]\frac{d^{2}u}{dt^{2}}[/tex] which equals X''Y
Or Ux for [tex]\frac{du}{dt}[/tex] which equals X'Y

But what about U'x

Is that a redundant way of saying the partial derivative of U with respect to x?

Or is it saying the derivative of the partial derivative of U with respect to x?

As I originally read it, I considered it X'Y, but now I'm wondering if maybe its X"Y.

Thanks!
 
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  • #2
frozenguy said:
Hey.. I ran across some problems and the notation used is a little different from what I've seen before.

considering U(x,y)=X(x)Y(y)
Sometimes I'll see Uxx for [tex]\frac{d^{2}u}{dt^{2}}[/tex] which equals X''Y
Or Ux for [tex]\frac{du}{dt}[/tex] which equals X'Y

I think you meant
Uxx for [tex]\frac{\partial^{2}u}{\partial x^{2}}[/tex]
or
Ux for [tex]\frac{\partial u}{\partial x}[/tex]

But what about U'x

Out of context, I would take that to mean [tex]\frac{d}{dt} \left(\frac{\partial u}{\partial x}\right)[/tex], but the context might indicate it meant something different. (The difference between [tex]d[/tex] and [tex]\partial[/tex] is not a typo.)
 
Last edited:
  • #3
AlephZero said:
I think you meant
Uxx for [tex]\frac{\partial^{2}u}{\partial x^{2}}[/tex]
or
Ux for [tex]\frac{\partial u}{\partial x}[/tex]



Out of context, I would take that to mean [tex]\frac{d}{dt} \left(\frac{\partial u}{\partial x}\right)[/tex], but the context might indicate it meant something different. (The difference between [tex]d[/tex] and [tex]\partial[/tex] is not a typo.)

Yes that was a bad typo I'm sorry. Glad you knew what I was talking about.

The context would be partial differential equations using separation of variables.

For example: [tex]U^{'}_{x}=U^{'}_{y}+U[/tex]
 
1.

What is the purpose of notation for separable partial differential equations?

The purpose of notation for separable partial differential equations is to provide a standardized way of writing and representing these types of equations. It allows for clear communication and understanding between mathematicians and scientists working with these equations.

2.

What is the difference between ordinary and partial differential equations?

The main difference between ordinary and partial differential equations is that ordinary differential equations involve only one independent variable, while partial differential equations involve multiple independent variables. This means that partial differential equations are used to model systems in which the dependent variable is a function of more than one independent variable.

3.

How do you determine if a partial differential equation is separable?

A partial differential equation is separable if it can be written as a product of two functions, one of which depends only on one independent variable and the other on the remaining independent variables. This means that the equation can be separated into simpler equations that can be solved independently.

4.

What are some common notations used in separable partial differential equations?

Some common notations used in separable partial differential equations include the use of subscripts to denote partial derivatives, the symbol ∂ to represent a partial derivative, and the use of Greek letters such as α, β, and γ to represent constants or coefficients.

5.

How is the solution to a separable partial differential equation written?

The solution to a separable partial differential equation is typically written in the form of a product of functions, one of which depends only on one independent variable and the other on the remaining independent variables. The constants or coefficients in the equation can also be included in the solution.

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