# Differential Eqns

1. Jan 18, 2005

### josephcollins

Can anyone offer some advice on this problem:

Obtain a general solution for the second order differential equation:

(d^2x/dx^2) - (dx/dt) - 2x = 10sint

I obtained the general solution and now need to determine the "solution which remains finite as t tends to infinity for which x=4 at t=0. Could anyone suggest how I may approach this

My general solution was:

G(x)= Ae^2x + Be^-x + cost - 3sint

Thanks for any help,
Joe

2. Jan 18, 2005

### dextercioby

Sorry,they have to have th same variable.It's either "t" or "x",make up your mind.Else it would have to be a PDE.

I think u can reject the positive exponential for obvious reasons...

Daniel...

3. Jan 18, 2005

### HallsofIvy

Well, one problem is you are confusing your dependent and independent variables!

x(t)= Ae2t+ Be-t+ cos(t)- 3 sin(t).

Now just substitute t= 0, x= 4 to get one equation in the two unknowns A and B.

Now what happens to e2t and e-t as t goes to infinity?
(sin(t) and cos(t) remain finite, of course). What do you need to do to make sure your solution doesn't go to infinity?

4. Jan 18, 2005

### josephcollins

ok, I have my general solution:

x: Ae^2t + Be^-t +cost -3sint

putting in x=4 and t=0 I obtain

4=A+B+1

so 3=A+B and A=3-B

So My final solution is:

x=(3-B)e^2t +Be^-t +cost -3sint

Is this correct, could someone verify? How about the fact that t tends to infinity, does this alter my answer at all???

5. Jan 18, 2005

### dextercioby

The problem specifically asks for the solution bounded at infinity.So that should give an idea about the value of B.

Daniel.

6. Jan 19, 2005

### josephcollins

The problem I have now is seeing what the equation does as t tends to infinity, e^-t will tend to zero, but e^2t will just tend to infinity while cost and sint are periodic, could you help me with this please?

Joe

7. Jan 19, 2005

### Justin Lazear

You've already found the problem and you already know what's going to happen as t goes to infinity. . e^(2t) is tending toward infinity as t goes to infinity. The problem specifies that the solution must remain finite as t goes to infinity, so you can't leave the e^(2t) in there, now can you? If you do, you're going to end up with a non-finite solution as t goes to infinity. What can you do with the coefficient B so that the problem term is no longer a factor (i.e. disappears)?

--J