What Are the Applications of Differential Equations in Mathematics?

[James...]

We did the basics of solving differential equations in Maths this week, well, it was actually just finding other approximate values of y for an equation given the derivative and a pair of co-ordinates on it.

Are these for equations that cannot be subject to intergration?

Also, what can they be applied to? I'm going to study Maths at University next year and have seen a lot of modules are based on differential equations, will this likely be using more accurate ways of solving the equation, or using it to explain things that are actually happening?

Our tutor wasn't very clued up on them and they got me interested.

Cheers
James

 

[LCKurtz]

Yes, many differential equations can not be explicitly solved, and numerical methods are used to approximate solutions of them.

Your second question has a very long answer. The short answer is they are used in almost any field that involves mathematics beyond calculus. Just a few examples are:

Equations of motion
Electric circuits
Mechanical systems
Finance
Thermodynamics

The full list would be very long.

 

[Bill_B]

I'm currently taking ODE in college, so let's see if my education is doing me any good

Let's say you have a derivative -

$$f'(x)=3x^{2}$$

That's easy enough to solve by integration --

$$\int f'(x) dx = \int 3x^2 dx$$

$$f(x) = x^3 + C$$

Notice in the original equation that $$f'(x)$$ is ONLY dependent on x.

Now look at this DE -

$$f'(x) = - f(x)$$

Here, $$f'(x)$$ is dependent on $$f(x)$$

If you integrate, you wind up with

$$f(x) = - \int f(x) dx$$

Which really doesn't help much. (well, I suppose you could guess at it for such a simple case).

Using other methods, you can actually solve for $$f(x)$$ in terms of x.

e.g., $$f(x)=e^{-x}$$Hopefully that helps a little. (and hopefully I didn't botch it too badly)

 

[elibj123]

Differential equations are not a tool to describe systems, differential equations are results of system analysis: once you analyze a system which changes in respect to its current state, you can have no other result but an ODE (or PDE).