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
Answer:
Explanation:
Given
Required
Find g(1)
If f(x) and g(x) are inverse functions, then:
implies that:
Using the analysis above:
implies that:
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