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ODE Linear System Complex Eigenvalues

  1. Dec 8, 2012 #1
    1. The problem statement, all variables and given/known data

    Solve the following systems by either substitution or elimination:

    dx/dt = y

    dy/dt = -x + cos(2t)

    2. Relevant equations

    I know the solution is:

    x(t) = c_1cos(t) + c_2sin(t) - 1/3cos(2t)

    y(t) = -c_1sin(t) + c_2cos(t) + 2/3sin(2t)

    3. The attempt at a solution

    x' = [ 0 1; -1 0][x; y] + cos(2t)[0; 1]

    Det(A-λI) = [-λ 1; -1 -λ] = λ^2+1 = λ_1 = i, λ_2 = -i

    λ = i; A-λi = [-i 1; -1 -i]

    (i)x + y = 0
    x = 1, y = -i;

    v = [1; -i] = [1; 0] + i[0; -1]

    x(t) = c_1*cos(t) + c_2*sin(t);

    y(t) = c_1*sin(t) - c_2*cos(t);

    [0 1; -1 0]*a = [0; -1]

    a = [1; 0]

    [0 1; -1 0]*b = [1; 0]

    b = [0; 1]

    x(t) = c_1*cos(t) + c_2*sin(t) + cos(2t);

    y(t) = c_1*sin(t) - c_2*cos(t) + 1;

    I used the Undetermined Coefficients method:

    http://tutorial.math.lamar.edu/Classes/DE/RealEigenvalues.aspx#Ex1_Start

    I don't understand what I'm doing wrong and I've tried using variation of parameters but I end up with a bunch of trig that I can't make anything out of. If someone can point out my error and help with deriving the problem correctly I would really appreciate it.
     
    Last edited: Dec 8, 2012
  2. jcsd
  3. Dec 8, 2012 #2

    Zondrina

    User Avatar
    Homework Helper

    So you have a non homogeneous system you want to solve. Let's denote the system [itex]x' = Ax + g(t)[/itex]

    The first thing you should do is solve the homogeneous system [itex]x' = Ax[/itex] by solving the characteristic polynomial [itex]p_A(λ)[/itex] for your eigenvalues and then proceed to find your eigenvectors. The eigenvectors will be easy since one is just a complex conjugate of the other.

    These will give you your fundamental homogeneous solution, lets call it [itex]x_c[/itex].

    Then you can construct a fundamental matrix [itex]ψ(t)[/itex] from the columns of your homogeneous solution.

    Then to solve the non homogeneous system for a particular solution [itex]x_p[/itex], I would recommend variation of parameters. So you should assume your particular solution has the form : [itex]x_p = ψ(t)u(t)[/itex] where [itex]u(t)[/itex] satisfies [itex]g(t) = ψ(t)u'(t)[/itex].

    After you solve [itex]g(t) = ψ(t)u'(t)[/itex] for [itex]u_{1}^{'}[/itex] and [itex]u_{2}^{'}[/itex], integrate them to find [itex]u_1[/itex] and [itex]u_2[/itex] which finally give you your vector [itex]u(t)[/itex]. Then simply do some matrix multiplication to find [itex]x_p[/itex].

    After solving for the homogeneous solution and the particular solution, your general solution will be [itex]x = x_c + x_p[/itex]

    I hope this helps you. It's a lot of work, but it's doable.
     
    Last edited: Dec 8, 2012
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