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Math Help - Differential equation

  1. #1
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    Differential equation

    I have a diff. equation given as;

    X{n+1} - \frac {{3}} {{2}}Xn + X{n-1} = n^{2} + 1

    n > 1

    What is the general solution
    X^{s}{n}
    to the assosiated diff. equation?

    Any help/pointers would be greatly appriciated!
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  2. #2
    Super Member Rebesques's Avatar
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    Re: Differential equation

    This is in fact the discrete analogue to a differential equation, called a difference equation.

    Begin by searching for solutions to the associated homogeneous equation: x_{n+1}-(3/2)x_n+x_{n-1}=0, n\in \mathbb{N}
    Set x_n=r^n, n\in \mathbb{N} and substitute to obtain the quadratic r^2-(3/2)r+1=0, with solutions
    r_{1,2}=\frac{3}{4}\pm{\rm i}\frac{\sqrt{7}}{4}=a\pm {\rm i}b.

    The general solution to the homogeneous equation becomes x_n=c_1a^n\cos(bn)+c_2a^n\sin(bn), n\in \mathbb{N}, where the c_i,i=1,2 are constants.

    Now we seek for a particular solution x_n^p to the original equation. Since the right hand side is a quadratic polynomial in n, we can try x_n^p=n^2+cn+d, where c,d are to be determined by substitution.

    After this is done, the general solution to the original equation is x_n=c_1a^n\cos(bn)+c_2a^n\sin(bn)+n^2+cn+d, n\in \mathbb{N}.
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