reduction of order
Sometimes you already know one solution of a second-order linear equation but need the second, independent one to complete the picture. Reduction of order is the technique that squeezes that second solution out of the first, turning a second-order problem into a first-order one — which is generally far easier.
Given one known solution y_1 of y'' + p(x) y' + q(x) y = 0, you look for a second solution of the form y_2 = v(x) y_1. Substituting and using that y_1 already solves the equation, all the v-without-derivative terms cancel, leaving a first-order equation in v' — the order has dropped by one. Solving it gives the standard formula y_2 = y_1 times integral of [ e^(-integral of p) / y_1^2 ]. The two solutions are automatically independent, so { y_1, y_2 } is a fundamental set.
Reduction of order is the workhorse behind several familiar results: it explains why a repeated characteristic root produces the extra factor of x (take y_1 = e^(r x) and the integral yields x e^(r x)), and it is the standard way to find the second solution of special-function equations once the first is known from a series. It also applies to certain nonlinear equations missing the dependent or independent variable, where substituting p = y' lowers the order directly.
x^2 y'' - 3x y' + 4y = 0 has the known solution y_1 = x^2 (a repeated root case). Reduction of order with y_2 = v x^2 leads to y_2 = x^2 ln x, completing the fundamental set { x^2, x^2 ln x }.
From one known solution, reduction of order manufactures a second, independent one.
Reduction of order needs the equation in the normalized form y'' + p y' + q y = 0 (leading coefficient 1) before reading off p; using the un-normalized p is a common source of a wrong second solution.