Reduction of order for linear ODEs

Reduction of order is a method used when solving a second-order linear ordinary differential equation (ODE) after one nontrivial solution is already known.

Suppose you are given a second-order linear ODE:

y^{″} + p (x) y^{'} + q (x) y = 0,

and a known solution $y_{1} (x)$ .
To find a second, independent solution $y_{2} (x)$ , you assume:

y_{2} (x) = v (x) y_{1} (x),

where $v (x)$ is an unknown function. Substituting into the ODE and simplifying leads to a first-order equation for $v^{'} (x)$ , which can be integrated to find $v (x)$ , and hence $y_{2} (x)$ .

This technique is crucial in the theory of linear differential equations to build a complete fundamental set of solutions.

It is the homogeneous sibling of variation of constants: both obtain a second solution by writing one with the shape of a known one and letting the multiplier depend on $x$ .