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