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Thread: Finding the Second Order Linear Homogeneous Equation from the given Fundamental Pair

  1. #1
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    Finding the Second Order Linear Homogeneous Equation from the given Fundamental Pair

    The problem instructions read like so:

    Show that the given functions are linearly independent on the interval I and find
    a second-order linear homogeneous equation having the pair as a fundamental set of
    solutions.

    $\displaystyle y_1=x$
    $\displaystyle y_2=x^2$
    $\displaystyle I=(\infty,-\infty)$

    I'm able to prove that they are linearly independent easily by simply using the Wronskian Matrix to get

    $\displaystyle W(x)=3x^3\neq0\in I$

    However, I don't know how to find the equation based off of what I'm given.
    I know the answer to the question (it's in the book) so I will post it for reference of what I'm looking for.

    $\displaystyle x^2 y\prime\prime - 2x y\prime +2y = 0$

    please show me the method to find this equation based off of the fundamental set given!
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  2. #2
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    Quote Originally Posted by Jenkins View Post
    The problem instructions read like so:

    Show that the given functions are linearly independent on the interval I and find
    a second-order linear homogeneous equation having the pair as a fundamental set of
    solutions.

    $\displaystyle y_1=x$
    $\displaystyle y_2=x^2$
    $\displaystyle I=(\infty,-\infty)$

    I'm able to prove that they are linearly independent easily by simply using the Wronskian Matrix to get

    $\displaystyle W(x)=3x^3\neq0\in I$

    However, I don't know how to find the equation based off of what I'm given.
    I know the answer to the question (it's in the book) so I will post it for reference of what I'm looking for.

    $\displaystyle x^2 y\prime\prime - 2x y\prime +2y = 0$

    please show me the method to find this equation based off of the fundamental set given!
    You should be able to set up three equations with three unknowns:

    $\displaystyle a_2(x)y''_1+a_1(x)y'_1+a_o(x)y_1=0$

    $\displaystyle a_2(x)y''_2+a_1(x)y'_2+a_o(x)y_2=0$

    $\displaystyle a_2(x)(y_1+y_2)''+a_1(x)(y_1+y_2)'+a_o(x)(y_1+y_2) =0$

    Try that...
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  3. #3
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    Quote Originally Posted by adkinsjr View Post
    You should be able to set up three equations with three unknowns:

    $\displaystyle a_2(x)y''_1+a_1(x)y'_1+a_o(x)y_1=0$

    $\displaystyle a_2(x)y''_2+a_1(x)y'_2+a_o(x)y_2=0$

    $\displaystyle a_2(x)(y_1+y_2)''+a_1(x)(y_1+y_2)'+a_o(x)(y_1+y_2) =0$

    Try that...
    I tried it and got that
    $\displaystyle a_2=0$
    $\displaystyle a_1=0$
    $\displaystyle a_o=0$

    which makes sense considering equations 1 and 2 both equal 0 and equation 3 is a linear combination of the two.

    Thanks for your help though
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  4. #4
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    Yeah, I was just about to edit that. It doesn't work.
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  5. #5
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    Ok, I think I know what to do now, lol. I was on the right track... Just define $\displaystyle P(x)=\frac{a_1(x)}{a_2(x)}$ and $\displaystyle Q(x)=\frac{a_o(x)}{a_2(x)}$, basically divide the first two equations (from my failed system) by the coefficient of y'' to get:

    $\displaystyle y_1''+P(x)y'_1+Q(x)y_1=0$

    $\displaystyle y_2''+P(x)y'_2+Q(x)y_2=0$

    From the first:

    $\displaystyle P(x)=-Q(x)x$

    Now substitute into the second:

    $\displaystyle 2-Q(x)2x^2+Q(x)x^2=0$

    $\displaystyle Q(x)=\frac{2}{x^2}$

    That should help...

    Remember we defined $\displaystyle Q(x)=\frac{a_o(x)}{a_2(x)}$.
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  6. #6
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    Quote Originally Posted by adkinsjr View Post
    Ok, I think I know what to do now, lol. I was on the right track... Just define $\displaystyle P(x)=\frac{a_1(x)}{a_2(x)}$ and $\displaystyle Q(x)=\frac{a_o(x)}{a_2(x)}$, basically divide the first two equations (from my failed system) by the coefficient of y'' to get:

    $\displaystyle y_1''+P(x)y'_1+Q(x)y_1=0$

    $\displaystyle y_2''+P(x)y'_2+Q(x)y_2=0$

    From the first:

    $\displaystyle P(x)=-Q(x)x$

    Now substitute into the second:

    $\displaystyle 2-Q(x)2x^2+Q(x)x^2=0$

    $\displaystyle Q(x)=\frac{2}{x^2}$

    That should help...

    Remember we defined $\displaystyle Q(x)=\frac{a_o(x)}{a_2(x)}$.
    you are magical
    Last edited by Jenkins; Feb 11th 2011 at 09:22 PM.
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