# LINEAR ALGEBRA: Given a sequence as a basis for a solution of some O.D.E., find it

1. Nov 16, 2006

### VinnyCee

Problem:

The sequence (c, s, 1, $$e_1,\,e_{-1}$$) is a basis for the solution space of some differential equation p(D)y = 0. Find this O.D.E.

NOTE: c = cos(t) and s = sin(t)

Work so far:

I know that $$e_1$$ gives a (t - 1) and that the $$e_{-1}$$ gives a (t + 1), but how do I solve for the 1! I think that the c and s give ($$t^2$$ - 1).

Also, can someone explain in detail or give a reference to what a Ker() is?

thanks

2. Nov 16, 2006

### 0rthodontist

Ker(x) means the kernel of the transformation x--the set of all values that x maps to the identity. In linear algebra that would be the set of all values that x maps to the zero vector. What exactly are $$e_1,\,e_{-1}$$?

3. Nov 17, 2006

### HallsofIvy

Staff Emeritus