y'(x) + x = y(x)
im told y(x)= sum(C_n)(x^n) power series method
i understand how to work the method im just confused about what happens to x?
does it go to 0?does it already have regular a singular point?help,thanks.
y'(x) + x = y(x)
im told y(x)= sum(C_n)(x^n) power series method
i understand how to work the method im just confused about what happens to x?
does it go to 0?does it already have regular a singular point?help,thanks.
See the attachment for the details but the x doesn't magically disappear
but the x term is incorporated into determining the coefficient of the x term once the power series and its derivative are plugged back into the differential equation
You'll notice c1-c0 = 0 considering the contstant term
then 2c2-c1+1 = 0 in considering the linear term this is where x comes in
There are no singularities in this DE
This is just y"- y= -x. x= 0 is not a singular point at all!
Let . Then so the equation becomes .
In order to be able to compare "like powers", change the indices. In the first sum, let j= n-2 so that n= j+2. We have
Setting coefficients of like powers equal we have
j= 0:
j= 1:
(That's where the "-x" on the right hand side comes in!)
j> 1: