Solving Power Series Problems: Finding 2 Solutions

[joker2014]

Hello.

I've been solving power series problems where the question asks to find 2 power series solutions.

I can solve it almost all, I can find the recurrence solution... however while checking the solutions, I see some answer solutions they used c0=1 and c1=0 to find the 2 solutions, and some answer solutions they did not use that.

am I missing something? I don't even understand why they used c0=1 c1=0, and when should i use or not.

 

[jedishrfu]

Could it have something to do with the function being even or odd?

 

[HallsofIvy]

It's not very clear what question you are asking! I think you are talking about a second order linear differential equation and you should understand that such equations will have two independent solutions such that all solutions can be written as a linear combination of those two.

In particular, given second order linear differential equation p(x)y''+ q(x)y'+ r(x)y= 0. In order to get a specific solution you would have to be give two additional conditions and the simplest would be the two initial value conditions, y(0)= 0, y'(0)= 1 and y(0)= 1, y'(0)= 0.

Of course, if we write the solutions in terms of a power series, $y(x)= a_0+ a_1x+ a_2x^2+ \cdot\cdot\cdot$ then $y'(x)= a_1+ 2a_2x+ 3a_3x^2+ \cdot\cdot\cdot$ so that $y(0)= a_0$ and $y'(0)= a_1$. So the simplest way to find two independent solutions, as power series, is to require that, in one, $a_0= 1$ and $a_1= 0$ while in the other $a_0= 0$ and $a_1= 1$.