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Homework Help: Are solultions of D.E. a Vector Space?

  1. Apr 21, 2006 #1
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    Let V be the solutions to the differential equation:

    [itex]
    a_{1}y' + a_{0} = x^2 + e^x
    [/itex]

    Decide using the properties of pointwise addition and scalar multiplication if V is a vector space or not.

    ---------------------

    Ok im having real trouble with this question. I can prove is something is a vector space aslong as I know what im trying to proove.


    To solve this question do I need to find or make up y? Or am I just assuming V = (a0, a1) and need to prove something like
    [tex]
    (a1 + b1)y' + (a0+b0) = x^2 + e^x + b1y' + b0
    [/tex]

    Could someone please help me get started? Thanks
     
  2. jcsd
  3. Apr 21, 2006 #2

    HallsofIvy

    User Avatar
    Science Advisor

    The fundamental property of a vector space is "linearity": if u and v are vectors, then so is au+ bv for any numbers a and b.

    Suppose y1 and y2 are solutions to that differential equation (that is they satisfy the equation) can you show that, for any number a and b, ay1+ by2 also satisfies the differential equation?
     
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