Reduction of order

Reduction of order is a process used to find the solution of a differential equation <math>y + p(t)y' + q(t)y = 0</math>, when Euler substitution methods find only one value for <math>\lambda</math>. (That is, <math>\sqrt {b^2-4ac} = 0</math> and <math>\lambda = \frac{-b}{2a}</math>.)

The process is carried out in the following manner:

1. <math>y_1 = e^{\lambda t}</math>. However, we must find <math>y_2</math>. <math>y_2</math> cannot be a multiple of <math>y_1</math>, so we assume that <math>y_2 = v(t) y_1</math>.

2. We apply the product rule to find:

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

4. We collect the <math>v(t)</math>, <math>v(t)'</math>, and <math>v(t)</math> terms:

5. The terms <math>(2y_1' + p(t) y_1)</math> and <math>(y_1 + p(t) y_1' + q(t) v(t) y_1)</math> equal zero in most instances, leading to the conclusion:

<math>v(t)(y_1) = 0</math>

Integrating twice, we yield:

6. This is enough to say that <math>y_2 = v(t) y_1 = ty_1 = t e^{\lambda t}</math>

7. The solution is then <math>y = c_1 y_1 + c_2 y_2 = e^{\lambda t} (c_1 + c_2 t)</math>

Reduction of order

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