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Linear homogenous ODEs with constant coefficients

  1. Nov 12, 2012 #1
    Given the ODE of the form:
    y''(x) + A*y'(x) + B*y(x) = 0

    If we choose a solution such that y(x) = e[itex]^{mx}[/itex]
    and plug it into the original ODE, the ODE becomes:
    (m[itex]^{2}[/itex] + A*m + B)e[itex]^{mx}[/itex] = 0

    If we solve for the roots of the characteristic equation such that
    m = r[itex]_{1}[/itex], r[itex]_{2}[/itex] (root 1 and root 2, respectively)

    The solution to the ODE would have the form:
    y(x) = c*e[itex]^{r_{1}*x}[/itex] + d*e[itex]^{r_{2}*x}[/itex], where c and d are constants

    My question is, why are the constants where they are in the solution? In other words, why are they multiplying y[itex]_{1}[/itex] & y[itex]_{2}[/itex], where
    y[itex]_{1}[/itex] = e[itex]^{r_{1}*x}[/itex] and y[itex]_{2}[/itex] = e[itex]^{r_{2}*x}[/itex] ?

    Why are the constants not just added to y(x), such that the solution to the ODE would be as follows, where k is a constant.
    y(x) = e[itex]^{r_{1}*x}[/itex] + e[itex]^{r_{2}*x}[/itex] + k

    This question mainly is for second order and higher differential equations. I understand how it works for first order linear homogenous DEs because the constant is simply the constant of integration, but I am having trouble understanding how it applies to higher orders.
     
    Last edited: Nov 12, 2012
  2. jcsd
  3. Nov 12, 2012 #2

    lurflurf

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    Homework Helper

    That is because the differential equation is linear. That is if L[y]=0 is a linear differential equation and u and v are any two solutions so that L=L[v]=0 then L[a u+a v]=0.
     
  4. Nov 13, 2012 #3

    HallsofIvy

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    Staff Emeritus
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    The fundamental theorem for such equations is:
    "The set of all solutions to a linear homogeneous differential equation of order n form a vector space of dimension n"

    That means that if we can find a set of n independent solutions, a basis for that vector space of solutions, any solution can be written as a linear combination of those solutions. And a "linear combination" means a sum of the functions multiplied by constants.
     
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