Homogenous linear DE with power series... value of constant?

[palex71]

Hello,
I have the following D.E. with y as a function of x:

y'' + (2/x) y' + y = 0

I went through the entire power series expansion and came up with the following:

y = c + 0x - (c/3!)x^2 + 0x^3 + (c/5!)x^4 + 0x^5 - (c/5!)x^6 + ...

I know the correct solution to be y=sinx/x, which is to what the series equates provided that c=1.
However, I am not seeing how I can justify positing c=1.

Any thoughts? Thanks!

 

[palex71]

Bumping... anyone have any insight?... Thanks!

 

[Ishuda]

Hello,
I have the following D.E. with y as a function of x:

y'' + (2/x) y' + y = 0

I went through the entire power series expansion and came up with the following:

y = c + 0x - (c/3!)x^2 + 0x^3 + (c/5!)x^4 + 0x^5 - (c/5!)x^6 + ...

I know the correct solution to be y=sinx/x, which is to what the series equates provided that c=1.
However, I am not seeing how I can justify positing c=1.

Any thoughts? Thanks!

Looking at your solution you have
y(x) = c sinc(x) = c sin(x) / x
which is A proper solution but might not be THE proper solution. Is this the complete question? Are there any boundary conditions?

 

[Bob Brown MSEE]

Complete Solution has two arbitrary constants

C₁ sin(x+C₂)/x

 

[palex71]

So sorry... i did find a relevant boundary condition that I hadn't applied. Thanks everyone!