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Math Help - Non-homogeneous 2nd order DE with constant coefficients.

  1. #1
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    Non-homogeneous 2nd order DE with constant coefficients.

    d^2y/dx^2-7 dy/dx +6y=36x
    given that when y=0    ,dy/dx=4 and x=0

    i would appreciate some help with this ones is all new to me. what do i do with the variables given.
    Last edited by mr fantastic; April 9th 2010 at 12:41 AM.
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  2. #2
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    It's a nonhomogeneous linear equation. So you'll need to find a particular solution of the nonhomogeneous equation and the general solution of the related homogeneous equation


    The homogeneous equation is  \frac{d^{2}y}{dx^{2}} - 7 \frac{dy}{dx} + 6y = 0

    The characteristic equation is  r^{2}-7x+6 =0 , which has roots of r=6 and r =1.

    So the general solution of the homogeneous equation is  y_{h}(x) = C_{1}e^{6x}+C_{2}e^{x}


    Now we need any particular solution of  \frac{d^{2}y}{dx^{2}} - 7 \frac{dy}{dx} + 6y = 36

     y_{p}(x)=6 will do


    so the general solution on the nonhomongeneous equation is  y(x) = y_{h}(x)+y_{p}(x) = C_{1}e^{6x}+C_{2}e^{x} + 6


    Now we need to find the two coefficients by using the initial conditions.

     y(0)=0=C_{1}+C_{2} + 6

     \frac{dy}{dx} = 6C_{1}e^{6x}+C_{2}e^{x}

     \frac{dy}{dx} (0) = 4=  6C_{1}+C_{2}

    Solving the two equations simultaneously,  C_{1}= 2 and  C_{2}= -8

    so our final solution is  y(x)= 2e^{6x}-8e^{x}+6
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  3. #3
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    thanks alot i was not familar with the solution for the general solution on the nonhomongeneous equation. thanks alot i really appreciate it.
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