1. ## 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 $\displaystyle (x^2+2x-1)y"-2(x+1)y'+2y=0$

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 $\displaystyle y=x^r$)

Any clarification to help get me started is appreciated. Thanks.

2. ## Re: Clarification of Problem

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

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

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

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

For example for the linear polynomial

$\displaystyle y(x)=ax+b$

$\displaystyle y'(x)=a$

$\displaystyle y''(x)=0$

so plugging these in we get

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

$\displaystyle -2ax-2a+2ax+2b=0$

$\displaystyle a=b$

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.