3 years, 3 months ago
What are differential equations and what kind of problems are they used for?
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M$2 Answers
They are used extensively in physics and engineering, as well, of course, by mathematicians. That's the easy part of the answer. Much more complicated is explaining what they are.
It's hard to understand what a Differential Equation is without first knowing a bit of Calculus. If you're asking about them in the first place, maybe you know what a derivative is already, but here's a brief explanation:
Suppose a function f(x). f'(a) (pronounced f prime of a) is the instantaneous rate of change of the function at the point a. f'(x) is the general rate of change at any point x in the function.
Consider f(x) = x^2, and let's say a = 3.
Then f(a) = 9, but what is the rate of change at point a in the function? Well, you're going to have to trust me here if you don't know derivatives already that f'(x) = 2x.
So f'(a) = 18. The rate of change at the point 3 of f(x) = x^2 is 18, and the general rate of change at any point x is 2x.
Okay, now for the really confusing stuff.
You can take a derivative of a derivative. So in the example above, f''(x) would be called the Second Order derivative of f(x). Again, take my word for it that it is 2. f'''(x) which is the Third Order derivative would be 0. But all you really need to take from this is that there are derivatives of increasing order, as you keep taking the derivative (called differentiating--that word seems familiar.) of a function over and over.
So what's a differential equation? Well, it's a function ultimately, so let's keep calling it f(x). But this function is expressed by the derivatives of various orders of another function. So we have our f(x), but if it's a differential equation, it'll look something like f(x) = f'(y) + f''(y) + f'''(y)... and so on.
And it just keeps getting more confusing from there, so I'll stop. But that's the basic idea. I hope it helps.
It's hard to understand what a Differential Equation is without first knowing a bit of Calculus. If you're asking about them in the first place, maybe you know what a derivative is already, but here's a brief explanation:
Suppose a function f(x). f'(a) (pronounced f prime of a) is the instantaneous rate of change of the function at the point a. f'(x) is the general rate of change at any point x in the function.
Consider f(x) = x^2, and let's say a = 3.
Then f(a) = 9, but what is the rate of change at point a in the function? Well, you're going to have to trust me here if you don't know derivatives already that f'(x) = 2x.
So f'(a) = 18. The rate of change at the point 3 of f(x) = x^2 is 18, and the general rate of change at any point x is 2x.
Okay, now for the really confusing stuff.
You can take a derivative of a derivative. So in the example above, f''(x) would be called the Second Order derivative of f(x). Again, take my word for it that it is 2. f'''(x) which is the Third Order derivative would be 0. But all you really need to take from this is that there are derivatives of increasing order, as you keep taking the derivative (called differentiating--that word seems familiar.) of a function over and over.
So what's a differential equation? Well, it's a function ultimately, so let's keep calling it f(x). But this function is expressed by the derivatives of various orders of another function. So we have our f(x), but if it's a differential equation, it'll look something like f(x) = f'(y) + f''(y) + f'''(y)... and so on.
And it just keeps getting more confusing from there, so I'll stop. But that's the basic idea. I hope it helps.
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M$
While @gonzojoe explained mathematically what a differential equation is (and very well, I may add), it may be a little more in detail for someone who has never taken physics or calculus.
Derivatives are great for determining one value based on a range of another factor. In real world calculations, the "other factor" that is probably most easily understood is time.
For example, speed is a derivative of distance. If someone is traveling at a constant speed, it's relatively easy to determine how much distance has been traveled over a certain amount of time. Speed x time = distance is the simplest kind of differential in this case because the speed is a constant and never changes.
Getting more difficult, you can throw in a constant acceleration. Since you're accelerating, the speed is constantly changing so if you're trying to figure out how far something has traveled, it becomes a lot more difficult because the distance you traveled in the first second is going to be less than the next second because the speed is constantly changing. A differential equation would allow you to find out how far you've driven after a given period of time.
Heck, you can use differential equations to determine other things too, like the amount of gas you have left in your tank only using things like fuel economy.
I hope this wasn't completely confusing and/or wrong. If only I could remember how to do them. I've not taken calculus in about 15 years and have completely forgotten how to do it. There are some things here at work I could use it for to satisfy my curiosity.
Derivatives are great for determining one value based on a range of another factor. In real world calculations, the "other factor" that is probably most easily understood is time.
For example, speed is a derivative of distance. If someone is traveling at a constant speed, it's relatively easy to determine how much distance has been traveled over a certain amount of time. Speed x time = distance is the simplest kind of differential in this case because the speed is a constant and never changes.
Getting more difficult, you can throw in a constant acceleration. Since you're accelerating, the speed is constantly changing so if you're trying to figure out how far something has traveled, it becomes a lot more difficult because the distance you traveled in the first second is going to be less than the next second because the speed is constantly changing. A differential equation would allow you to find out how far you've driven after a given period of time.
Heck, you can use differential equations to determine other things too, like the amount of gas you have left in your tank only using things like fuel economy.
I hope this wasn't completely confusing and/or wrong. If only I could remember how to do them. I've not taken calculus in about 15 years and have completely forgotten how to do it. There are some things here at work I could use it for to satisfy my curiosity.
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M$
In rereading my answer, let me just say that I noticed a small error. Where I said f'(a) = 18, it actually should have been f'(a) = 6. It doesn't really effect the rest of my answer, but being a stickler for the details is important. It was a dumb error like mine here that crashed the old mars lander.
Well I'm in the process of taking AP Calculus AB, so I'm well familiar with derivatives and integrals.
What I'm trying to figure out is exactly what a "differential equation" and when to use them.
Cochese did a fine job of explaining their applications. Basically, they are used when you are trying to find out something that depends on the derivatives of other functions, which is usually just a fancy way of saying that you're trying to calculate something based on several elements that are constantly changing.
If you're in AP Calc, I'll go ahead and assume you've had some physics and you've done the old cannonball parabola trace. It is absolutely true that, according to Newtonian physics, a cannonball shot up at an angle against gravity will trace a perfect section of a parabola, and you can express that with simple algebra.
But there's a big difference between the real world and that Ideal world where there's only the cannonball and gravity. When we start to consider wind resistance, irregularities in the cannonball's mass, irregularities in the barrel of the cannon that shot it, and probably a few other factors I'm not thinking of, then we can no longer use simple algebra to predict its path, because it depends on too many other things. When these other things can be expressed as derivatives of another function, then we can use a differential equation to describe a more precise path of the cannonball.
I hope this and Cochise's answer have made things a little more clear.