Math Is Fun Forum

Daniel123

Member

Registered: 2007-05-23

Posts: 663

Second order differential equations

I've just moved on to second order (linear) differential equations.

My book goes through the 'proof' that the general solution of the second order differential equation,

,

whose auxiliary quadratic has distinct, real roots

and

, is given by:

where A and B are arbitrary constants.

It shows this by assuming that there are two distinct solutions of the differential equation u and v (which are either constants or functions of x), stating that the general solution is given by

, and showing that if this is the case, the differential equation is in fact satisfied (by writing

etc is terms of A, B, u and v). It then goes on to say that specific solutions are found by trying

, differentiating, substituting, and factorising to find the roots of the auxiliary equation (I'm skimming through this as I assume you are familair with this 'proof').

Now I understand each of these steps, but I'm not really sure why they decided that y = Au + Bv would satisfy the equation in the first place? I can kind of see it, but I'm not 100%. Where did this come from? Does the formal proof assume this, or does it show it?

Also, why must the general solution contain e? Again, I can kind of see why (mainly due to e's importance in integration), but could someone explain fully? Obviously there will be no e (as alpha and beta will be 0) if there is only a second derivative in the equation, but otherwise it seems there will always be one/two?

Also, the book says that "when solving a second order differential equation, the general solution will have two arbitrary constants". Is this just because you are effectively integrating twice?

Thanks

Last edited by Daniel123 (2008-06-16 02:39:35)

gnitsuk

Member

Registered: 2006-02-09

Posts: 121

Re: Second order differential equations

Hi, hope this helps a bit:

So we have:

Without loss of generaility and using the more compact D-operator notation our starting point is:

Now suppose the auxiliary equation has real roots 

and 

then we have:

Then the operator has corresponding factors and so we can write our initial equation as:

Let

so that

Now this is a first order differential equation and so we solve it using first order methods to obtain

So there is the origin of the exponential function in our solution.

Thus we have:

That is:

There are two cases:

Case 1:

The solution is:

and so

Case 2:

Then:

Giving

That is:

And yes, there will be two constants precisely because we are integrating twice.

Last edited by gnitsuk (2008-06-16 03:41:27)