Definition of a linear differential equation

In summary, the conversation is discussing the definition of a partial linear differential equation and whether the use of products in the equation is allowed for it to still be considered linear. The definition given states that a linear differential equation cannot have products of the function and/or its derivatives and that the function and its partial derivatives can only occur to the first degree. There is some concern about the use of "and/or" in the definition and a question about the validity of including products in the equation. Another question is posed about the homogeneity and linearity of an equation if a function multiplied by u is involved.
  • #1
Niles
1,866
0

Homework Statement


Hi all. Is the following definition of a partial linear differential equation correct?


A partial differential equation is called linear if no products of the function and/or its derivatives occur, and if the unknown function and its partial derivatives occur only to the first degree.


I am a little worried about the "and/or" regarding products in the DE. Is it correct that the following term is NOT allowed in a linear differential equation?:

[tex]
\frac{\partial u}{\partial x}\frac{\partial u}{\partial t}
[/tex]

- and can you confirm the validity of the "and/or" in my definition? Thanks in advance.

sincerely,
Niles.
 
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  • #2
Can anybody confirm this?

And also: If I have a function f(x,y)*u, then is the equation still homogeneous and linear? As an example, take f(x,y)=cos(x):
 

What is a linear differential equation?

A linear differential equation is a mathematical equation that describes the relationship between a function and its derivatives. It can be written in the form of y'(x) = f(x), where y is the dependent variable, x is the independent variable, and y' is the first derivative of y. It is called "linear" because the dependent variable and its derivatives appear only in a linear form.

What makes a differential equation linear?

A differential equation is considered linear if the dependent variable and its derivatives appear only in a linear form, meaning they are raised to the first power and are not multiplied together or divided by each other. Nonlinear terms, such as y2 or y*y', would make the equation nonlinear.

What is the general solution to a linear differential equation?

The general solution to a linear differential equation is a formula that satisfies the equation for all possible values of the independent variable. It includes a constant called the arbitrary constant that can take on any value. This constant allows for an infinite number of solutions to the equation.

What is the order of a linear differential equation?

The order of a linear differential equation is the highest derivative that appears in the equation. For example, y'' + 2y' + 3y = 0 is a second-order linear differential equation because it contains the second derivative y''. The order of the equation determines the number of arbitrary constants in the general solution.

What are some real-life applications of linear differential equations?

Linear differential equations are used to model many real-world phenomena, such as population growth, chemical reactions, and electrical circuits. They are also commonly used in physics, engineering, and economics to describe the relationships between variables and their rates of change.

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