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Math Help - differential equations with constant coefficients

  1. #1
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    differential equations with constant coefficients

    Please help me with this one:
    y''-9y'+20y=x^{2}e^{4x}
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  2. #2
    Eater of Worlds
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    Lets use Variation of Parameters. I have always preferred that method, but that's me.

    From the auxiliary equation we get:

    m^{2}-9m+20=(m-5)(m-4), \;\ m=5, \;\ m=4

    y_{c}=C_{1}e^{4x}+C_{2}e^{5x}

    Now use the determinants to find W, W1, W2:

    W=\begin{vmatrix}y_{1}&y_{2}\\y_{1}^{'}&y_{2}^{'}\  end{vmatrix}=\begin{vmatrix}e^{4x}&e^{5x}\\4e^{4x}  &5e^{5x}\end{vmatrix}=e^{9x}

    W_{1}=\begin{vmatrix}0&y_{2}\\f(x)&y_{2}^{'}\end{v  matrix}=\begin{vmatrix}0&e^{5x}\\x^{2}e^{4x}&5e^{5  x}\end{vmatrix}=-x^{2}e^{9x}

    W_{2}=\begin{vmatrix}y_{1}&0\\y_{1}^{'}&f(x)\end{v  matrix}=\begin{vmatrix}e^{4x}&0\\4e^{4x}&x^{2}e^{4  x}\end{vmatrix}=x^{2}e^{8x}

    u_{1}^{'}=\frac{W_{1}}{W}=-x^{2}

    u_{2}^{'}=\frac{W_{2}}{W}=\frac{x^{2}}{e^{x}}

    Integrate to find u1 and u2:

    u_{1}=-\int x^{2}dx=\frac{-1}{3}x^{3}

    u_{2}=\int\frac{x^{2}}{e^{x}}dx=\frac{-x^{2}-2x-2}{e^{x}}

    y_{p}=u_{1}+u_{2}

    \boxed{y=y_{c}+y_{p}=\left(\frac{-1}{3}x^{3}-x^{2}-2x-2\right)e^{4x}+C_{1}e^{4x}+C_{2}e^{5x}}

    There is a nice outline for future reference.
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  3. #3
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    Thanks a lot!
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