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Basis of the vector space of solutions to a differential equation.

  1. Mar 5, 2009 #1
    Apologies, have solved this question.
    Answer if useful for anyone:
    Basis= {e^(5x),e^(-10x)}.

    1. The problem statement, all variables and given/known data

    Consider the differential equation
    (2nd derivative of y wrt x) + 5(1st derivative of y wrt x) - 50y =0

    Find a basis of the vector space of solutions of the above differential equation. You should separate each vector with a comma, and each should take the value of 1 at x=0

    2. Relevant equations

    -

    3. The attempt at a solution

    Solution: y(x)=Ae^(5x) + Be^(-10x)
    The solutions are a linear combination of these two terms.
    I'm unsure of how to convert this into 'a basis of the vector space of solutions'.
     
    Last edited: Mar 5, 2009
  2. jcsd
  3. Mar 5, 2009 #2

    lurflurf

    User Avatar
    Homework Helper

    So take values of A and B, to give particular solutions. List them.
    The value must not be the same and A+B must equal one.
    say s!=t
    y1=s*exp(5x)+(1-s)*exp(-10x)
    y2=t*exp(5x)+(1-t)*exp(-10x)

    Basis={y1,y2}={s*exp(5x)+(1-s)*exp(-10x),t*exp(5x)+(1-t)*exp(-10x)}
    you can also just chose particular s and t
    s=10,t=101
    Basis={y1,y2}={10*exp(5x)-9*exp(-10x),101*exp(5x)-100*exp(-10x)}
    The book answer results from the choice s=1,t=0
     
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