Reduction of order

Reduction of order (or d’Alembert reduction) is a technique in mathematics for solving second-order linear ordinary differential equations. It is employed when one solution ${\displaystyle y_{1}(x)}$ is known and a second linearly independent solution ${\displaystyle y_{2}(x)}$ is desired. The method also applies to n-th order equations. In this case the ansatz will yield an (n−1)-th order equation for ${\displaystyle v}$.