I think you meant to write
which is an ODE of Cauchy-Euler type. Let . Then
Substituting and its derivatives into the ODE gives
Divide through by , which we can do because
:
Since this root has multiplicity 2, we get the characteristic solution
If you're not sure where the logarithm comes from, scroll to the bottom for a bit more in-depth explanation.
With the given initial values, we find
so that the particular solution is
# # #
Under the hood, we're actually substituting , so that
. When we do this, we need to account for the derivative of
wrt the new variable
. By the chain rule,
Since is a function of
, we can treat
in the same way, so denote this by
. By the quotient rule,
and by the chain rule,
where
so that
Plug all this into the original ODE to get a new one that is linear in with constant coefficients:
which has characteristic equation
and admits the characteristic solution
Finally replace to get the solution we found earlier,