Differential equations y''-xy'+2y=0?

[hawaiidude]

how would you solve

y''-xy'+2y=0?

it says, {2a2+6a3x+12a4x^2+...n(n-1_anx^n-1

ahh i don't know..how would you solve this? can someone explain? my math teacher is un willing to help me...

 

[HallsofIvy]

You mean your teacher is unwilling to tell you the answer. How mean of him! Does he expect you to actually LEARN how to do it?

The problem you have is a linear homogeneous d.e. with variable coefficients. The standard way of solving such an equation is to use power series (as you can see from the given solution).

Let y= Σ a_nxⁿ where the a_n are to be determined. Calculate the derivatives of that, plug into your equation and combine like powers of x. You will get a recurrence relation for the a_ns.

 

[hawaiidude]

yeah..thnaks but why are there two types of reserection and when do i use them? like sigma n=0 anX^n and other one with the & lambda ; x^ sigma anX^n...when do i use which ones and how? and how do i plug in answers in the differential equation/ like 8x^2y''+10xy'+(x-1)y=0
x^ &lambda ; {8 &lambda ; -1) a0 +10 &lambda ; ...and so on...any help is welcomed

 

[HallsofIvy]

Please do not post the same thing on different threads. I responded to this on a different thread.