# Reduction of order

### From Conservapedia

**Reduction of order** is a process used to find the solution of a differential equation *y*'' + *p*(*t*)*y*' + *q*(*t*)*y* = 0, when Euler substitution methods find only one value for λ. (That is, and .)

The process is carried out in the following manner:

1. *y*_{1} = *e*^{λt}. However, we must find *y*_{2}. *y*_{2} cannot be a multiple of *y*_{1}, so we assume that *y*_{2} = *v*(*t*)*y*_{1}.

2. We apply the product rule to find:

*y*_{2} = *v*(*t*)*y*_{1}

*y*_{2}' = *v*(*t*)'*y*_{1} + *v*(*t*)*y*_{1}'

*y*_{2}'' = *v*(*t*)''*y*_{1} + 2*v*(*t*)'*y*_{1}' + *v*(*t*)*y*_{1}''

3. We substitute these expressions into the initial differential equation:

*y*_{2}'' + *p*(*t*)*y*_{2}' + *q*(*t*)*y* = (*v*(*t*)''*y*_{1} + 2*v*(*t*)'*y*_{1}' + *v*(*t*)*y*_{1}'') + *p*(*t*)(*v*(*t*)'*y*_{1} + *v*(*t*)*y*_{1}') + *q*(*t*)(*v*(*t*)*y*_{1})

4. We collect the *v*(*t*)'', *v*(*t*)', and *v*(*t*) terms:

*v*(*t*)''(*y*_{1}) + *v*(*t*)'(2*y*_{1}' + *p*(*t*)*y*_{1}) + *v*(*t*)(*y*_{1}'' + *p*(*t*)*y*_{1}' + *q*(*t*)*v*(*t*)*y*_{1})

5. The terms (2*y*_{1}' + *p*(*t*)*y*_{1}) and (*y*_{1}'' + *p*(*t*)*y*_{1}' + *q*(*t*)*v*(*t*)*y*_{1}) equal zero in most instances, leading to the conclusion:

*v*(*t*)''(*y*_{1}) = 0

Integrating twice, we yield:

*v*(*t*) = *c*_{1}*t* + *c*_{2}

6. This is enough to say that *y*_{2} = *v*(*t*)*y*_{1} = *t**y*_{1} = *t**e*^{λt}

7. The solution is then *y* = *c*_{1}*y*_{1} + *c*_{2}*y*_{2} = *e*^{λt}(*c*_{1} + *c*_{2}*t*)