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2nd order non homogeneous diff. eqs. 2nd posting for clarification

  1. Mar 27, 2010 #1
    I recently attempted to solve the following:

    y” + (K/m)y = (Kl[tex]^{0}[/tex]+mg)/m

    y(0) = l[tex]_{0}[/tex]

    y(t[tex]_{e}[/tex]) = (K l[tex]_{0}[/tex]+mg)/K

    3. The attempt at a solution

    y(t) = -(mg/K)cos{[tex]\sqrt{K/m}[/tex] t} + (mg/K){cos{[tex]\sqrt{K/m}[/tex] t[tex]_{e}[/tex]}/sin{[tex]\sqrt{K/m}[/tex] t[tex]_{e}[/tex]}}*sin{[tex]\sqrt{K/m}[/tex] t} + (K l[tex]_{0}[/tex]+mg)/K

    which works. But the auxiliary equation has imaginary roots and one of the texts I'm reading states that

    y(t) = (C[tex]_{1}[/tex] + C[tex]_{2}[/tex])cos{[tex]\sqrt{K/m}[/tex] t} + i(C[tex]_{1}[/tex] - C[tex]_{2}[/tex])sin{[tex]\sqrt{K/m}[/tex] t} + (K l[tex]_{0}[/tex]+mg)/K

    should be a solution w/imaginary coefficients. I get

    C[tex]_{1}[/tex] = -mg/K - C[tex]_{2}[/tex]

    C[tex]_{2}[/tex] = (-i/2)(mg/K)(cos{[tex]\sqrt{K/m}[/tex] t[tex]_{e}[/tex]}/sin{[tex]\sqrt{K/m}[/tex] t[tex]_{e}[/tex]}

    but they don't solve the original condition reqs. What am I doing wrong/not seeing?

    P.S. Why don't some of the Latex scripts work from the pull down menu; ie subscript gives superscript?

    I posted this earlier, but cannot retrieve the posting. I'm still having trouble with the Latex formatting. Subscript gives superscript for some reason. And I'm using the full down menu option and inserting the arg. into the script. (?) Also, regarding the coefficients, please see the following, Case III:

    http://www.stewartcalculus.com/data...ntexts/upfiles/3c3-2ndOrderLinearEqns_Stu.pdf
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Mar 27, 2010 #2

    Mark44

    Staff: Mentor

    Let's take a look at the homogeneous equation,
    y'' + a2y = 0

    You can write the solutions in either of two ways:
    y = c1eiax + c2e-iax

    or

    y = c1cos(ax) + c2sin(ax)

    The reason you can do this is that eiax = cos(ax) + i sin(ax), and e-iax = cos(ax) - i sin(ax), from which you can isolate the real part from the imaginary part like so:
    (1/2)(eiax + e-iax) = cos(ax)
    (1/(2i))(eiax - e-iax) = sin(ax)

    So both cos(ax) and sin(ax) are linear combinations of eiax and e-iax.

    To answer your question about the initial condition, your particular solution should look like this:

    [tex]y_p(x) = C[/tex]
    Plug that into your differential equation to solve for C.

    Your general solution should look like this:
    [tex]y(x) = c_1 cos(bx) + c_2 sin(bx) + C[/tex]
    where C has been solved for previously.

    To find c1 and c2, use your initial conditions
    [tex]y(0) = l_0[/tex]
    and
    y(te) = (Kl0 + mg)/K

    Regarding your question about LaTeX subscripts, they are working fine for me, using both the [ sub] style not inside [ tex] tags, and using the _<subscript> style inside [ tex] tags. The problem is probably that you are using [ tex] tags in isolation when it would be better to surround the entire equation with them.

    IOW -- don't do this l [ tex]_0[ /tex]. Instead, do this -- [ tex]y(0) = l_0[ /tex]
    Note that I have left a leading space at the start of the tex tags so that they won't render.
     
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