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General Solution of DE

  1. Oct 21, 2009 #1
    1. The problem statement, all variables and given/known data

    Verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. Form the solution.

    [tex]4y^{''} - 4y^{'} + y = 0[/tex]

    [tex]e^{x/2}, xe^{x/2}[/tex]

    2. Relevant equations

    [tex]Wr(y_1,...,y_n)=det\left(\begin{array}{ccc}y_1&\cdots&y_n\\y_1\prime&\cdots&y_n\prime\\\vdots&\vdots &\vdots\\y_1^{(n-1)}&\cdots&y_n^{(n-1)}\end{array}\right)[/tex]

    3. The attempt at a solution

    [tex]e^{x/2}, xe^{x/2}[/tex]

    [tex]\frac{e^{x/2}}{2}, \frac{xe^{x/2}}{2} + e^{x/2}[/tex]

    [tex]Wr(e^{x/2}, xe^{x/2})= det\left(\begin{array}{ccc}e^{x/2}&xe^{x/2}\\\frac{e^{x/2}}{2}&\frac{xe^{x/2}}{2} + e^{x/2}\\\end{array}\right)[/tex]

    [tex](e^{x/2})(\frac{xe^{x/2}}{2} + e^{x/2}) - (xe^{x/2})(\frac{e^{x/2}}{2})[/tex]

    [tex]\frac{xe^{x/2}}{2} + e^{x} - \frac{xe^{x/2}}{2} = e^{x} \neq 0[/tex]

    Therefore the functions are linearly independent and form a solution.


    [tex]y = c_{1}e^{x/2} + c_{2}xe^{x/2}[/tex]
     
  2. jcsd
  3. Oct 21, 2009 #2

    LCKurtz

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    I don't see a question. Your work looks correct as far as it goes. But have you shown that your two functions are actually solutions?
     
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