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Linear 2nd order diff. eqn.

  1. Oct 7, 2008 #1
    1. This is exercise 3.19.15 from Boyce & DiPrima's Differential Equations.

    [tex] u'' + u = F(t), u(0) = 0, u'(0) = 0 [/tex]

    [tex] 0 \leq t < \pi , F(t) = F_0t [/tex]
    [tex] \pi \leq t \leq 2\pi , F(t) = F_0(2\pi-t) [/tex]
    [tex] t > 2\pi , F(t) = 0 [/tex]

    Solve for [tex] u(t) [/tex].



    2. The solution to the homogenous equation is

    [tex] u'' + u = 0 [/tex]

    is [tex] u(t) = c_1 cos(t) + c_2 sin(t) [/tex].



    3. Plugging in the boundary conditions yields

    [tex] u(0) = c_1 + 0 = 0 [/tex]
    [tex] u'(t) = - c_1 sin(t) + c_2 cos(t) [/tex]
    [tex] u'(0) = c_2 = 0 [/tex].

    Then a particular solution of the form [tex] u_p = A t [/tex] works

    [tex] u'' + u = At = F_0t [/tex],

    and so the solution for [tex] 0 \leq t < \pi [/tex] I find to be [tex] u_1(t) = F_0 t [/tex]. But the solution in the back of the textbook states [tex] u_1(t) = F_0t - F_0

    sin(t) [/tex], which clearly also satisfies the differential equation and the initial value conditions, but it's inconsistent with what I get. Where did I go wrong?
     
  2. jcsd
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