Can a Differential Equation Have an Unrelated Variable?

[MathewsMD]

I'm not sure if this is particularly important, but so far through my studies I've only encountered DE with two related variables (e.g. $\frac {dy}{dx} = 3x$).

Now, given another function with an additional variable that is UNRELATED to the two other variables, can this still be considered a differential equation (e.g. $\frac {dy}{dx} = 3x + z$ where z is a random variable)? Does this not meet it's definition?

If I'm not mistaken, all the variables have to be related and it is possible to have DE with infinite variables, as long as they are all related. Is my understanding wrong?

 

[Mark44]

I'm not sure if this is particularly important, but so far through my studies I've only encountered DE with two related variables (e.g. $\frac {dy}{dx} = 3x$).

Now, given another function with an additional variable that is UNRELATED to the two other variables, can this still be considered a differential equation (e.g. $\frac {dy}{dx} = 3x + z$ where z is a random variable)? Does this not meet it's definition?

I would say no. Your first equation above is an example of an ordinary differential equation, in which you typically have one variable as a function of another. The assumption is that y = f(x), for some unknown function of a single independent variable.

Another type of differential equation is the partial differential equation, or PDE, in which you typically have a dependent variable that is a function of two or more independent variables, and the equation to be solved is some combination of partial derivatives of the unknown function.

The notation dy/dx implies that y is a function of x alone. In a PDE you have partial derivatives of various orders, such as $\frac{\partial f}{\partial x}$ and $\frac{\partial^2 f}{\partial x^2}$ and so on.

When you said that z is a random variable, you are probably unaware that "random variable" is a term widely used in statistics, with a specific meaning.

If I'm not mistaken, all the variables have to be related and it is possible to have DE with infinite variables, as long as they are all related. Is my understanding wrong?

 

[micromass]

I'm not sure if this is particularly important, but so far through my studies I've only encountered DE with two related variables (e.g. $\frac {dy}{dx} = 3x$).

Now, given another function with an additional variable that is UNRELATED to the two other variables, can this still be considered a differential equation (e.g. $\frac {dy}{dx} = 3x + z$ where z is a random variable)? Does this not meet it's definition?

If I'm not mistaken, all the variables have to be related and it is possible to have DE with infinite variables, as long as they are all related. Is my understanding wrong?

You can always solve the equation $\frac{dy}{dx} = 3x + z$ by treating $z$ as a constant (if it doesn't depend on $x$). But we don't call $z$ a variable then, but rather a parameter.