Suppose that one solution y1(x) of the homogeneous second-order linear differential equation
y″ + p(x)y′ + q(x)y = 0
is known (on an interval I where p and q are continuous functions). The method of reduction of order consists of substituting y2(x) = u(x)y1(x) in (18) and attempting to determine the function u(x) so that y2(x) is a second linearly independent solution of (18). After substituting y = u(x)y1(x) in Eq. (18), use the fact that y1 (x) is a solution to deduce that
y1v″ + (2y1′ + py1v′ = 0.
If y1(x) is known, then (19) is a separable equation that is readily solved for the derivative v′(x) of y(x). Integration of y′(x) then gives the desired (nonconstant) function u(x).
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