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July 7th, 2013, 12:22 AM   #1
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Is this an ordinary diff eq or a partial diff eq?

What if we have a diff eq where the unknown function depends on one variable (let's say u) and also explicitly on time (t)

i.e. the unknown function = x(u,t)

Is time really another variable, making the diff eq a partial diff eq? I've heard this explicit dependence on t by x be called "non-autonomous." Not sure if these terms are mutually exclusive or not.

What if u also depends on time? What if u does not depend on time?

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July 7th, 2013, 06:36 AM   #2
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Re: Is this an ordinary diff eq or a partial diff eq?

Time is a variable if it's a variable. If, say, y is the temperature on a bar (with x the length along the bar), and y = f(x), then f'(x) is the temperature gradient along the bar and this a normal derivative.

If, however, you consider how the bar is cooling over time, then y = f(x, t). f is now a function of two variables, so the derivative above becomes a partial derivative of f (wrt x). This describes the temperature along the bar at a given point in time. And, you have another partial derivative of f which is wrt t. This describes the change in temperature over time at a given point along the bar.
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July 8th, 2013, 10:01 AM   #3
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Re: Is this an ordinary diff eq or a partial diff eq?

I think what you are saying is that you know how y depends upon t and have a differential equation showing how it depends upon x. In that case, we would consider t a parameter and have an ordinary differential equation for y as a function of x.
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