When looking up at the clouds, you can usually make out different shapes and figures with _____.
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Each of these ODEs is linear and homogeneous with constant coefficients, so we only need to find the roots to their respective characteristic equations.
(a) The characteristic equation for
is
which arises from the ansatz .
The characteristic roots are and
. Then the general solution is
where are arbitrary constants.
(b) The characteristic equation here is
with a root at of multiplicity 2. Then the general solution is
(c) The characteristic equation is
with roots at , where
. Then the general solution is
Recall Euler's identity,
Then we can rewrite the solution as
or even more simply as