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Math Help - Clarification of Problem

  1. #1
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    Clarification of Problem

    I have just completed a section on Second ODE and Second Order Linear Homogenous equations. Topics were the superposition principle, the wronskian, and linear dependence and independence.

    But Part A of this exercise question is not clear to me.

    Given the diff equ (x^2+2x-1)y"-2(x+1)y'+2y=0

    a) Show that the equation has a linear polynomial and a quadratic polynomial as solutions. ??
    b) Find two linearly independent solutions of the equation and find the general solution. (For this do i assume the answer is in the form y=x^r)


    Any clarification to help get me started is appreciated. Thanks.
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  2. #2
    MHF Contributor
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    Re: Clarification of Problem

    looking at various solutions I'm pretty sure you mean (x^2+2x+1)y'', not what you have written.

    a linear polynomial is pl(x)=ax+b

    a quadratic polynomial is pq(x)=a x^2+b x + c

    first step should just be plugging these in and confirming that they are solutions.

    For example for the linear polynomial

    y(x)=ax+b

    y'(x)=a

    y''(x)=0

    so plugging these in we get

    (x^2+2x+1) \cdot 0 - 2(x+1)\cdot a + 2(ax+b)=0

    -2ax-2a+2ax+2b=0

    a=b

    so any linear polynomial of the form ax+a is a solution. This can be thought of as a(x+1), i.e. (x+1) serves as a basis vector of the solution space.

    Now repeat this with the quadratic. and remember superposition, and then you'll start to understand what they are after in part 2 of the problem.
    Last edited by romsek; February 11th 2014 at 09:35 PM.
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