# Second order linear differential operator

1. Aug 5, 2009

1. The problem statement, all variables and given/known data
Suppose that L is a second order linear differential operator over the interval J, that f is a function defined on J, and that the function v has the property that

Lv = f on J

(a) Show that if y = u + v and that Lu = 0 on J, then Ly = f on J
(b) Show that if Ly = f on J, then y = u + v for some u such that Lu = 0 on J
2. Relevant equations
None that apply
3. The attempt at a solution
Well unfortunately this one I was unable to attempt because i am not even sure what it is asking, this professor I have tends to deviate from the book.

2. Aug 5, 2009

### hatsoff

(a) If L is linear, then L(u+v)=Lu+Lv. So if we have Lu=0, then

Ly=L(u+v)=Lu+Lv=0+Lv=Lv=f.

(b) Since L is linear, then there is a zero vector u with Lu=0 and 0+y=y. Choose u=0 and v=y. Then u+v=0+y=y.

3. Aug 5, 2009