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Is “ordinary differential equations in more than two variables” correct? Used in two books

I need an example of a differential equation (nontrivial) that involves more than one unknown functionsEliminate the arbitrary funcion - PDE first orderFinding a general solution of a partial differential equation.Didactic examples in linear ordinary differential equationsUsing the two partial differential equations construct a partial differential equation containing only variable 'G' and solve it.Geometric proof of uniqueness and existence of ordinary differential equationsSolving Differential Equations without separation of variablesWhat is main differences between these Differential Equations books?How to differentiate a given unknown function involving three variables and two arbitrary constants?Use the method of separation of variables to separate the differential equation into 3 ordinary differential equations.

3

$begingroup$

I have seen the formulation "ordinary differential equations in more than two variables" in two books.

Is this terminology really correct?

The books:

The first book is Elements of Partial Differential Equations by Ian N. Sneddon. Chapter 1 is called Ordinary differential equations in more than two variables
(the chapter at Google Books)
:

In this chapter we shall discuss the propertires of ordinary differential equations in more than two variables.

...

... if in the rectangular Cartesian coordinates $(x,y,z)$ of a point in three-dimensional space are connected by a single relation of the type $$ f(x,y,z)=0 tag1$$
the point lies on a surface.

...

To demonstrate this generally we suppose a point $(x,y,z)$ satisfying equation $(1)$. Then any increments $(delta x, delta y, delta z)$ in $(x,y,z)$ are related by the equation $$fracpartial fpartial xdelta x + fracpartial fpartial ydelta y +fracpartial fpartial zdelta z=0$$
so that two of them can be chose arbitrarily.

The second book is A Treatise on Differential Equations by George Boole and chapter XII is called Ordinary differential equations in more than two variables (the chapter at Google Books):

The class of equations which we shall first consider in this Chapter, is represented by the typical form,
$$
P, dx + Q, dy + R, dz = 0,
$$
$P$, $Q$ and $R$ being functions of the variables $x$, $y$, $z$; and it is usually termed a total differential equation of the first order with three variables.

I'm confused over the terminology of using "ordinary" and "many variables" in the same sentence. Isn't a ordinary differential equation always one variable and if more than one variable it's instead a partial differential equation?

ordinary-differential-equations pde terminology

share|cite|improve this question

edited Mar 22 at 14:07

JDoeDoe

asked Mar 22 at 13:40

JDoeDoeJDoeDoe

7851615

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  They are Partial Differential Equations (PDE). Don't be confused by the terminology. The important is de definition given specifically in each book even if the symbols and terms are not always standard, especially in old littérature.
  $endgroup$
  – JJacquelin
  Mar 22 at 14:33
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  In the case of the book by Boole, I would say that there has been a shift in the meaning of "ordinary" and "partial" differential equations between then and now. But the book by Sneddon is recent, so I cannot explain his use of the terminology. Perhaps by reading the books, the meaning would become clear.
  $endgroup$
  – Paul Sinclair
  Mar 22 at 23:42
  

  
  
  
  

add a comment | 

3

$begingroup$

I have seen the formulation "ordinary differential equations in more than two variables" in two books.

Is this terminology really correct?

The books:

The first book is Elements of Partial Differential Equations by Ian N. Sneddon. Chapter 1 is called Ordinary differential equations in more than two variables
(the chapter at Google Books)
:

In this chapter we shall discuss the propertires of ordinary differential equations in more than two variables.

...

... if in the rectangular Cartesian coordinates $(x,y,z)$ of a point in three-dimensional space are connected by a single relation of the type $$ f(x,y,z)=0 tag1$$
the point lies on a surface.

...

To demonstrate this generally we suppose a point $(x,y,z)$ satisfying equation $(1)$. Then any increments $(delta x, delta y, delta z)$ in $(x,y,z)$ are related by the equation $$fracpartial fpartial xdelta x + fracpartial fpartial ydelta y +fracpartial fpartial zdelta z=0$$
so that two of them can be chose arbitrarily.

The second book is A Treatise on Differential Equations by George Boole and chapter XII is called Ordinary differential equations in more than two variables (the chapter at Google Books):

The class of equations which we shall first consider in this Chapter, is represented by the typical form,
$$
P, dx + Q, dy + R, dz = 0,
$$
$P$, $Q$ and $R$ being functions of the variables $x$, $y$, $z$; and it is usually termed a total differential equation of the first order with three variables.

I'm confused over the terminology of using "ordinary" and "many variables" in the same sentence. Isn't a ordinary differential equation always one variable and if more than one variable it's instead a partial differential equation?

ordinary-differential-equations pde terminology

share|cite|improve this question

edited Mar 22 at 14:07

JDoeDoe

asked Mar 22 at 13:40

JDoeDoeJDoeDoe

7851615

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  They are Partial Differential Equations (PDE). Don't be confused by the terminology. The important is de definition given specifically in each book even if the symbols and terms are not always standard, especially in old littérature.
  $endgroup$
  – JJacquelin
  Mar 22 at 14:33
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  In the case of the book by Boole, I would say that there has been a shift in the meaning of "ordinary" and "partial" differential equations between then and now. But the book by Sneddon is recent, so I cannot explain his use of the terminology. Perhaps by reading the books, the meaning would become clear.
  $endgroup$
  – Paul Sinclair
  Mar 22 at 23:42
  

  
  
  
  

add a comment | 

$begingroup$

I have seen the formulation "ordinary differential equations in more than two variables" in two books.

Is this terminology really correct?

The books:

The first book is Elements of Partial Differential Equations by Ian N. Sneddon. Chapter 1 is called Ordinary differential equations in more than two variables
(the chapter at Google Books)
:

In this chapter we shall discuss the propertires of ordinary differential equations in more than two variables.

...

... if in the rectangular Cartesian coordinates $(x,y,z)$ of a point in three-dimensional space are connected by a single relation of the type $$ f(x,y,z)=0 tag1$$
the point lies on a surface.

...

To demonstrate this generally we suppose a point $(x,y,z)$ satisfying equation $(1)$. Then any increments $(delta x, delta y, delta z)$ in $(x,y,z)$ are related by the equation $$fracpartial fpartial xdelta x + fracpartial fpartial ydelta y +fracpartial fpartial zdelta z=0$$
so that two of them can be chose arbitrarily.

The second book is A Treatise on Differential Equations by George Boole and chapter XII is called Ordinary differential equations in more than two variables (the chapter at Google Books):

The class of equations which we shall first consider in this Chapter, is represented by the typical form,
$$
P, dx + Q, dy + R, dz = 0,
$$
$P$, $Q$ and $R$ being functions of the variables $x$, $y$, $z$; and it is usually termed a total differential equation of the first order with three variables.

I'm confused over the terminology of using "ordinary" and "many variables" in the same sentence. Isn't a ordinary differential equation always one variable and if more than one variable it's instead a partial differential equation?

ordinary-differential-equations pde terminology

share|cite|improve this question

edited Mar 22 at 14:07

JDoeDoe

asked Mar 22 at 13:40

JDoeDoeJDoeDoe

7851615

$endgroup$

share|cite|improve this question

edited Mar 22 at 14:07

JDoeDoe

asked Mar 22 at 13:40

JDoeDoeJDoeDoe

7851615

share|cite|improve this question

edited Mar 22 at 14:07

JDoeDoe

asked Mar 22 at 13:40

JDoeDoeJDoeDoe

7851615

asked Mar 22 at 13:40

JDoeDoeJDoeDoe

7851615

asked Mar 22 at 13:40

JDoeDoeJDoeDoe

7851615