Can't understand differential equations?

Question

I am having a hard time understanding the format and how differential equations are written and what they are exactly. I understand derivatives and integrals pretty well from Calc 1 and some of Calc II but Diff Eq's I am not getting, the syntax is weird and I am having trouble with homework!. Any simple sites or explanations of first order Diff Eq's and how to solve them? (Wikipedia is not a simple explanation or site!)

Answer 1

check the following

(1) http://mathworld.wolfram.com/First-Order...

(2) http://www.sosmath.com/diffeq/first/firs...

Answer 2

The first step to solve is to seperate the variables with respective d-variables. Then take integral so simple

Answer 3

Let's leave the issue of solving differential equations alone for the moment-- it's a large subject and whole books can be written about it-- but understanding them is actually quite simple.

Think first about "ordinary" equations, for example:

x^2 - 9 = 0

You can put any number you like into the expression on the left, but the equation is only true for certain very specific values (namely, 3 and -3).

A differential equation is the same idea but there's a critical difference: instead of trying to find a _number_ to satisfy the equality, you're trying to find a _function_ to satisfy the equality.

Here's another really simple example:

y - cos(x) = 0

Here now y is a function, y(x), rather than just a number.  And it should be pretty obvious I think that y = cos(x) satisfies that equation.  Now that was a pretty trivial example but let's make it a bit trickier:

dy/dx - cos(x) = 0

Now we're looking for a function once again, y(x), such that if you take the derivative of the function and then subtract cos(x) you get 0.  You can try yourself and see that y = sin(x) satisfies that equation (in fact, y = sin(x) + any constant also works!).  That there is the basic idea of a differential equation.  The hard part, in general, is finding the right y(x) to satisfy the equation.

Often times you'll see the unknown function y showing up more than once in an equation.  For example:

dy/dx - 3y = 0

Here it's maybe not so easy to just guess the solution.  It turns out a solution is y(x) = e^(3x).  (Again there's more than one solution.  y = Ce^(3x) is a solution for any constant value C.)

Sometimes differential equations can be really complex.  For example,

dy/dx + 3y^4 + sin(y) - cos(x) = 0

Don't ask me to solve that!  It may not even be possible.  But even if you can't solve it, you can understand what it means.  You're looking for a function such that if you take its derivative, then add 3 times the function to the fourth power, then add the sine of the function, and then subtract cos(x), you get 0.  I have no idea what function satisfies that equation-- probably there isn't one-- but if someone came to you and claimed to have a solution, you could check it easily by just plugging their solution into that equation and seeing if it's true.

So that's the basic idea behind the meaning of a DE.  They're important because they arise naturally in the description of a lot of important problems.  And like I said, solving them is the hard part and could occupy many volumes.  Hope that gets you started though!