Reduction of Order4 For a solution y1 of
y" + p(x)y' + q(x)y = 0
On interval I. such that y1 is not the zero function on I, use the following steps to find the conditions on a function v of x such that
y2 = uy1
Is a solution to equation (16) that is linearly independent from y1 on I.
(a) Determine y'2 and y''2 and substitute them into equation (16.) Regroup and use the fact that y1 is a solution of (16) to obtain
y1v" + (2y'1 + py1)v' = 0.
(b) Set v' = w. Solve the resulting first-order DE to obtain

So that

(c) Establish the fact that {y1, y2} is a linearly independent set by showing that v cannot be a constant function on I. Show that v' cannot be identically zero on I.

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