Reduction of order
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Reduction of order is a process used to find the solution of a differential equation 
, when Euler substitution methods find only one value for 
. (That is, 
and 
.)
The process is carried out in the following manner:
1. 
. However, we must find 
. cannot be a multiple of 
, so we assume that 
.
2. We apply the product rule to find:
3. We substitute these expressions into the initial differential equation:
4. We collect the 
, 
, and 
terms:
5. The terms 
and 
equal zero in most instances, leading to the conclusion:
Integrating twice, we yield:
6. This is enough to say that 
7. The solution is then 
