Covered today: (NOTE: some of this is not in the book) The basics of linear operators. The differential
operator and using it to solve th-order constant coefficient DEs.
For next time:
Use the differential operator to solve the DE
Use the differential operator to solve the DE
Consider the DE . Find a solution to this DE which is
a constant function. Use that solution, along with Theorem 3 in Section 18.2 and your
solution on the problem above, to find the general solution to the DE
.
Solve the DE using the differential operator. Compare
with your solution from the previous problem.
Solve the following nonhomogeneous DEs using the differential operator
.
.
Suppose the polynomial has real roots and
and that . Prove that the general solution to the DE
is
.
Suppose the polynomial has a repeated root . Prove
that the general solution to the DE
is
.
Suppose and are linear operators. Define a new linear
operator by
. Prove that is in
fact a linear operator. Be careful to justify all your steps.
Let be a linear operator, and let be a
function.
Define an
operator by . Prove that is in fact a linear
operator.