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For this case, the first thing to do is to graph the following ordered pairs:
(-6, -1)
(-3, 2)
(-1,4)
(2,7)
We observe that the graph is a linear function with the following equation:
y = x + 5
Note: see attached image.
Answer:
The function that best represents the ordered pairs is:
y = x + 5
First look for the fundamental solutions by solving the homogeneous version of the ODE:

The characteristic equation is

with roots
and
, giving the two solutions
and
.
For the non-homogeneous version, you can exploit the superposition principle and consider one term from the right side at a time.

Assume the ansatz solution,



(You could include a constant term <em>f</em> here, but it would get absorbed by the first solution
anyway.)
Substitute these into the ODE:




is already accounted for, so assume an ansatz of the form



Substitute into the ODE:





Assume an ansatz solution



Substitute into the ODE:



So, the general solution of the original ODE is
