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Math Help - general solution to differential equation

  1. #1
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    general solution to differential equation

    For our practice final one of the questions is

    Find the general solution to the following differential equation: y'' - 4y' + 4y =x2 - 3x + 2

    I understand that one would add the particular solution and general solution. For the particular solution I got 1/4x2-1/4x+1/8 and for the general solution I got Ae2x+Be2x. However he gave us the solution and it was Ae-2x+Bxe-2x+1/4x2-1/4x+1/8. Does anyone see where I went wrong? I can show more work if thats helpful.
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  2. #2
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    Re: general solution to differential equation

    Quote Originally Posted by JML2618 View Post
    For our practice final one of the questions is

    Find the general solution to the following differential equation: y'' - 4y' + 4y =x2 - 3x + 2

    I understand that one would add the particular solution and general solution. For the particular solution I got 1/4x2-1/4x+1/8 and for the general solution I got Ae2x+Be2x. However he gave us the solution and it was Ae-2x+Bxe-2x+1/4x2-1/4x+1/8. Does anyone see where I went wrong? I can show more work if thats helpful.
    Your mistake was in not dealing with the fact that your characteristic polynomial has a single root of order 2.
    That makes your homogeneous solution of the form

    y[x]=c_1e^{2x}+c_2xe^{2x} (note the factor x in the 2nd term)

    go use this form for your homogeneous solution and resolve your particular solution.
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    Re: general solution to differential equation

    oh okay i understand now thanks for the reply! Do you use this whenever it's in the form y''+y'+y and ce^ax +ce^ax when it's y''+y'?
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    Re: general solution to differential equation

    or is it that the roots are equal?
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    Re: general solution to differential equation

    Quote Originally Posted by JML2618 View Post
    or is it that the roots are equal?
    Pauls Online Notes : Differential Equations - Repeated Roots
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    Re: general solution to differential equation

    Notice that Ae^{2x}+ Be^{2x}= Ce^{2x} with C= A+ B. Those are not two independent solutions and cannot give the general solution to a second order equation.
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