Solving fourth order differential equation ( )

  1. Solving fourth order differential equation (URGENT)

    I have two second order differential equation which needs to be solved.

    x1''(t) = 8 x2(t)
    x2''(t) = 2 x1(t)

    I have the initial conditions, x1(0) = 0, x2(0) = 1, and terminal conditions x1(pi/4) = 1, x2(pi/4) = 0.

    Can anyone help me solve these equations?? What I did was to write the equations in terms of x1 and x2 respectively, but that gives me a fourth order differential

    x1''''(t) = 16 x1(t)
    x2''''(t) = 16 x2(t)

    and I do not know how to solve for these. Can any one help please??
     
  2. jcsd
  3. Cyosis

    Cyosis 1,495
    Homework Helper

    Re: Solving fourth order differential equation (URGENT)

    You know that the exponential, sine and cosine return to their original form after differentiating them four times. So try a solution of the form [itex]c_1 e^{a t}+c_2 e^{-at}+c_3 \sin(at)+c_4 \cos(at)[/itex].
     
  4. Re: Solving fourth order differential equation (URGENT)

    So the solution would be
    x1(t) = c1 e^(16t) + c2 e^-(16t) + c3 sin(16t) + c4 cos(16t)??
     
  5. Cyosis

    Cyosis 1,495
    Homework Helper

    Re: Solving fourth order differential equation (URGENT)

    No you found the wrong a. Differentiating e^16t 4 times would give 16^4 e^16t, which is not a solution.
     
  6. Re: Solving fourth order differential equation (URGENT)

    would a = 4 be the correct solution??
     
  7. Cyosis

    Cyosis 1,495
    Homework Helper

    Re: Solving fourth order differential equation (URGENT)

    I am not sure why you have to ask. Take the derivative of your function with a=4 , four times and you will see that it is not the correct solution.
     
  8. Re: Solving fourth order differential equation (URGENT)

    I am sorry, I was taking the second dervivative for some reason. I conclude that a = 2 would be thr right solution. Am I correct??
     
  9. Cyosis

    Cyosis 1,495
    Homework Helper

    Re: Solving fourth order differential equation (URGENT)

    Yes that's correct.
     
  10. Re: Solving fourth order differential equation (URGENT)

    In order to solve for c1,c2,c3 and c4, I would need the second differential of x1. Would that be x1'' = c1 e^4t +c2 e^-4t ??
     
  11. Re: Solving fourth order differential equation (URGENT)

    Your equations are linear with constant coefficients. I would handle either the following ways

    1. Apply the Laplace transform to the equations - this will transform the problem to solving algebraic system of equations or

    2. Let the solution be x1(t)=q1ert and x2(t)=q2ert . Substitute this assumption and you can determine r, q1 and q2 from eigenvalue-eigenvector equation.
     
  12. Re: Solving fourth order differential equation (URGENT)

    I got the solution to the equation using the fourth order differntial, but am stuck wolving for the constants c1,c2,c3,c4. If I wanted the second order differntail for x1, would that be x1'' = c1 e^4t +c2 e^-4t ??
     
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