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Homework Help: Differential Equations Homework

  1. Mar 11, 2010 #1
    1. The problem statement, all variables and given/known data

    1. Let D and I be the differential operator and the identity operator, respectively. Find two
    real-valued functions f(t) and g(t) such that:
    D[tex]^{2}[/tex] + I = (D + f(t)I)(D + g(t)I):

    2. Use this theorem to prove the corollary given below.
    Theorem: There are two nonzero solutions y[tex]_{1}[/tex] and y[tex]_{2}[/tex] to the differential equation
    y''+ p(t)y' + q(t) = 0
    such that one of the two functions is not a constant multiple of the other, and
    that c[tex]_{2}[/tex]y[tex]_{1}[/tex]+c[tex]_{2}[/tex]y[tex]_{2}[/tex] for arbitrary constants c[tex]_{1}[/tex] and c[tex]_{2}[/tex] is a general solution to the
    differential equation.
    Corollary: If z[tex]_{1}[/tex] and z[tex]_{2}[/tex] are two nonzero solutions to the differential equation such that one of the two functions is not a constant multiple of the other, then c[tex]_{1}[/tex]z[tex]_{1}[/tex] + c[tex]_{2}[/tex]z[tex]_{2}[/tex] for arbitrary constants c[tex]_{1}[/tex] and c[tex]_{2}[/tex] is a general solution the
    differential equation.

    3. Prove the theorem stated in #2.


    2. Relevant equations

    1. I know there is a theorem such that ((D[tex]^{2}[/tex] + a[tex]^{2}[/tex]I) = 0 and then this equation equals c[tex]_{1}[/tex]cos(at) + c[tex]_{2}[/tex]sin(at)


    3. The attempt at a solution

    1. I tried to do this, but got lost. I know that
    (D[tex]^{2}[/tex] + I) = c[tex]_{1}[/tex]cos(1t) + c[tex]_{2}[/tex]sin(1t).
    That's all I've got.

    2. My teacher assigned this problem, but I have no clue how to go about proving this corollary. The corollary seems like the same as the theorem except with using z's instead of y's and i'm confused.

    3. Again, I have no clue how to prove this theorem, so I need some help.


    Thanks guys!
     
  2. jcsd
  3. Mar 11, 2010 #2
    For some reason, it didn't subscript the c's and y's and z's. All of those are supposed to be written as subscripts, not superscripts. The only superscripts are just the D's and they are squared.
     
  4. Mar 11, 2010 #3

    HallsofIvy

    User Avatar
    Science Advisor

    Any thing inside the [ tex ] tags tends to get put above the line anyway.

    Put the whole thing inside [ tex ]: [ tex ]c_2[ /tex ]: [tex]c_2[/tex], not c[ tex ]_2[ /tex]: c[tex]_2[/tex].

    In fact, it is better to put the whole formula inside the tex tags.


    This problem has nothing to do with solving the equation.

    You are asked to find two functions, f(x) and g(x) so that
    (D+ f(x)I)(D+ g(x)I)= D^2+ I

    Go ahead and "multiply" the left side: (D+ f(x)I)(D+ g(x)I)= D(D+ g(x)I)+ f(x)I(D+ g(x)I)
    = D^2+ D(g(x))I+ f(x)D+ f(x)g(x)I= D^2+ g'(x)I+ f(x)D+ f(x)g(x)I.

    For that to be the same as D^2+ I, you must have f(x)= 0 and g'(x)+ f(x)g(x)= 1. The first of those equations makes it particularly easy!
     
    Last edited by a moderator: Mar 11, 2010
  5. Mar 11, 2010 #4
    Thank you so much!! That seems so easy, but I don't think I would have ever known that method, but now it makes so much sense. Thank you!

    Any input on the other 2 problems?
     
  6. Mar 11, 2010 #5
    My teacher gave the following hint:
    The theorem states the existence of such two functions, and we don't know what they are. What we know is that there ARE two such functions. That's what the theorem is about, namely, the existence of such two functions. Now, not knowing what those are, we pull out tricks and managed to find two solutions; for example, z1=t² and z2=t³. Using this example, I raise a question: are we going to reach all the solutions by considering
    a t² + b t³
    and all constants a and b?
     
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