Solving 2nd Order Diff EQ: G(x), Joe Needs Help

[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

 

[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...

 

[HallsofIvy]

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

x(t)= Ae^2t+ 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 e^2t 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?

 

[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?

 

[dextercioby]

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

Daniel.

 

[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

 

[Justin Lazear]

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

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