Try this solution (see the attachment); it consists of 5 basic steps.
The answers are marked with red and green colour.
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With reduction of order, we assume a solution of the form 
, with 
. Then
and substituting into the ODE gives
Let
, so that 
. This gives the linear ODE
This equation is also separable, so you can write
Integrating both sides with respect to
gives
Next, solve
for 
by integrating both sides again with respect to .
And finally, solve for
.
and note that
is already taken into account as part of , so this is the general solution to the ODE.
and substituting into the ODE gives
Let
This equation is also separable, so you can write
Integrating both sides with respect to
Next, solve
And finally, solve for
and note that