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Complex solutions to a differential equation a vector space?

  1. Oct 6, 2009 #1
    1. The problem statement, all variables and given/known data

    Is the set of all complex solutions to the differential equation [tex] \frac{d^2 y}{d x^2} + 2\frac{d y}{d x} - 3 y = 0[/tex]

    If so, find a basis, the dimension, and give the zero vector

    2. Relevant equations



    3. The attempt at a solution

    I solved the equation and got the answer:

    [tex]y(x) = C_1 e^{-3x} + C_2e^x[/tex]

    I know how to test if a set is a vector space but I'm not really seeing the "set" here. Is it because [tex]C_1[/tex] and [tex]C_2[/tex] can be complex numbers? In which case, wouldn't any complex number work so would I get the set of all complex numbers?

    any help is appreciated
     
  2. jcsd
  3. Oct 6, 2009 #2

    lanedance

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    i think you missed out somthing in your question -

    I'm assuming it is, "is the set of all complex solutions to the differential equation - a vector space"

    I would start with the axioms for a vector space - what are they?
    Then what is the general form of your solution? This will generally have some undetermined constants to give a family of solutions

    "the space" is then the set of all solutions. Is it a vector space?

    In short, a vector space is closed under scalar multiplication and addition, with some other axioms, so check:
    closure under scalar multiplication - so given any solution is a scalar times the solution also a solution
    closure under addition - so given 2 solutions is their sum also a solution

    then fill out the other axioms
     
    Last edited: Oct 6, 2009
  4. Oct 6, 2009 #3

    lanedance

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    updated post above
     
  5. Oct 6, 2009 #4

    HallsofIvy

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    First, exactly what do you mean by "complex solution"? If you mean simply that the coefficients C1 and C2 are complex, as you say, they the "set" asked about is NOT "the set of all complex numbers". It is the set of all such functions:
    [tex]\{ f(x)= C_1 e^{-3x}+ C_2 e^x : C_1, C_2 \in \math{C}\}[/tex].

    If you add two such functions is the sum also a function of that kind? If you multiply such a function by a complex number is the product also a function of that kind?
     
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