Reduction of order

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, $\sqrt {b^2-4ac} = 0$ and $\lambda = \frac{-b}{2a}$.)

The process is carried out in the following manner:

1. y1 = eλt. However, we must find y2. y2 cannot be a multiple of y1, so we assume that y2 = v(t)y1.

2. We apply the product rule to find:

y2 = v(t)y1

y2' = v(t)'y1 + v(t)y1'

y2'' = v(t)''y1 + 2v(t)'y1' + v(t)y1''

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

y2'' + p(t)y2' + q(t)y = (v(t)''y1 + 2v(t)'y1' + v(t)y1'') + p(t)(v(t)'y1 + v(t)y1') + q(t)(v(t)y1)

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

v(t)''(y1) + v(t)'(2y1' + p(t)y1) + v(t)(y1'' + p(t)y1' + q(t)v(t)y1)

5. The terms (2y1' + p(t)y1) and (y1'' + p(t)y1' + q(t)v(t)y1) equal zero in most instances, leading to the conclusion:

v(t)''(y1) = 0

Integrating twice, we yield:

v(t) = c1t + c2

6. This is enough to say that y2 = v(t)y1 = ty1 = teλt

7. The solution is then y = c1y1 + c2y2 = eλt(c1 + c2t)