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Differential Eqns

  1. Jan 18, 2005 #1
    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. jcsd
  3. Jan 18, 2005 #2

    dextercioby

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    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...
     
  4. Jan 18, 2005 #3

    HallsofIvy

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    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?
     
  5. Jan 18, 2005 #4
    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???
     
  6. Jan 18, 2005 #5

    dextercioby

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

    Daniel.
     
  7. Jan 19, 2005 #6
    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
     
  8. Jan 19, 2005 #7

    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
     
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