TABLE OF CONTENTS In Chapter 11 we discussed the general theory of second order linear equations. In the intervening chapters we have concentrated on linear equations with constant coefficients, but we now return to the more general case in which the coefficients are allowed to be functions of t,
We saw in Chapter 11 that in order fully to solve a second order homogeneous linear differential equation we need two linearly independent solutions. In this chapter we show that if we happen to know, or can guess, one solution of an equation like (17.1) then there is a systematic way to find a second, linearly independent, solution.
The method is called reduction of order , since it enables us to use our knowledge of one solution to find a first order differential equation that we can use to find the second solution.
Suppose we know that u( t) solves the second order linear equation
The idea is to make the substitution x( t) = u( t) y( t) and then solve the resulting equation for y( t). From x( t) = u( t) y( t) it follows that
and substituting these into the original equation gives
The terms in which the factor of y is not differentiated,
