Chapter 4.2, Problem 27P

Prove Theorem 4.2.4 and Corollary 4.2.5.

Reduction of Order. Given one solution y, of a second order linear homogeneous equation,

$y"+p(t)y\text{'}+q(t)y=0,$ $(i)$

a systematic procedure for deriving a second solution $y$, such that $\{y_1,y_2\}$ is a fundamental set is known as the method of reduction of order.

To find a second solution, assume a solution of the form $y=v(t)y_1(t)$. Substituting $y=v(t)y_1(t)$, $y\text{'}=v\text{'}(t)y_1(t)+v(t)y_1\text{'}(t)$ and $y"=v"(t)y_1(t)+2v\text{'}(t)y_1\text{'}(t)+v(t)y_1"(t)$ in Eq. $(i)$ and collecting terms give

$y_{1}v"+(2y_1\text{'}+p{y}_1)v\text{'}+(y_1"+p{y}_1\text{'}+q{y}_1)v=0.$ $(ii)$

Since $y_1$, is a solution of Eq. $(i)$, the coefficient of $v$ in Eq. $(ii)$ is zero, so that Eq. $(ii)$ reduces to

$y_{1}v"+(2y_1"+p{y}_1)v\text{'}=0$, $(iii)$

a first order equation for the function $w=v\text{'}$ that can be solved either as a first order linear equation or as a separable equation. Once $v\text{'}$ has been found, then $v$ is obtained by integrating

$w$ and then $y$ is determined from $y=v(t)y_1(t)$

This procedure is called the method of reduction of order, because the crucial step is the solution of a first order differential equation for $v\text{'}$ rather than the original second order equation for $y$.